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Never  before  appeared  in  a  School  Reader. 


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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 

Early  American  Mathematics  Books 


4- 


+- 


AN 


Elementary 


ARITHMETIC 


BY 

D.   B.   HAGAE,  Ph.D., 

PRINCIPAL  OF  STATE  NORMAL,  SCHOOL,  SALEM,  MASS. 


PHILADELPHIA 
COWPERTHWAIT    &   Co. 


HAGAR'S 

Mathematical  Series. 


I.  Hagar's  Primary  Lessons  in  Numbers. 

II.  Hagar's  Elementary  Arithmetic. 

III.  Hagar's  Common  School  Arithmetic. 

IV.  Hagar's  Elementary  Algebra. 


FOR    TEACHERS. 

Dictation  Problems  and  Reviews  .  .  $  .50 
Key  to  Common  School  Arithmetic  .  1.00 
Key  to  Elementary  Algebra  .     .     .     .     1.25 

Forwarded,  postpaid,  on  receipt  of  the  Price. 


Entered,  according  to  Act  of  Congress,  in  the  year  1871,  by 

DANIEL  B.  HAGAR   and  HENRY  B.  MAGLATHLIN, 
In  the  Office  of  the  L  ibrarian  of  Congress,  at  Washington. 


Copyright,  1877,  by 
DAN/EL   B.  HAGAR  and  HENRY  B.  MAGLATHLIN. 


Westcott    &    Thomson,  E.   Stanley  Hart, 

Stcr--   'ypers  and  Electrotypers,  Philada.  Printer,  Philada. 


INTRODUCTION. 


The  purpose  of  this  manual  is  to  facilitate  the  advance 
of  young  learners  in  the  science  of  numbers  by  gradual 
steps. 

The  lessons  are  intended  to  secure  that  normal  develop- 
ment and  discipline  of  the  reasoning  powers,  and  those 
correct  habits  of  investigation,  which  alone  form  a  sure 
foundation  for  progress  in  any  branch  of  knowledge. 

The  principles  and  rules  have  been  carefully  established 
by  induction.  The  plan  has  been  to  make  the  reasons  for 
each  process  entirely  clear,  and  to  enable  the  learner  to 
state  them  in  concise  language. 

Mental  and  written  exercises  admitting  substantially  of 
the  same  solution  have  been  combined,  so  as  to  render  un- 
necessary the  use  of  a  separate  mental  arithmetic,  and 
otherwise  to  abridge  advantageously  the  ordinary  course  of 
arithmetical  study. 

Pictorial  illustrations,  from  original  designs,  have  been 
freely  introduced,  with  the  view  of  making  some  parts  of 
the  subjects  treated  more  easily  understood,  through  the 
medium  of  the  eye. 

It  is  hoped  that  this  work,  which  is  complete  in  itself, 
may  satisfactorily  meet  the  wants  of  intermediate  classes 
in  graded  schools ;  and  also  may  prove  useful  in  many  dis- 
trict schools,  in  which  the  attendance  is  too  limited  to  war- 
rant the  use  of  a  more  extended  treatise. 

3 


SUGGESTIONS   TO  TEACHEES. 


1.  Material  objects  should  be  used  as  illustrations  far  enough  to 
make  sure  that  the  pupils  clearly  understand  the  value  of  numbers; 
how  numbers  are  composed  in  addition,  how  they  are  separated  in 
subtraction,  how  multiplication  and  division  are  performed,  and  how 
the  elementary  tables  are  constructed.  When  these  things  are  com- 
prehended, material  objects  should  be  dispensed  with,  and  a  thorough 
knowledge  of  the  tables  should  be  relied  upon  for  the  requisite  results. 

2.  The  tables  should  be  made  so  familiar  that  when  any  two  num- 
bers are  named,  the  result  of  a  desired  operation  upon  them  shall,  by 
the  power  of  repeated  association,  instantly  flash  upon  the  mind. 

3.  Care  should  be  taken  that  the  definitions  are  clearly  understood 
before  they  are  learned. 

4.  The  attention  of  pupils  should  be  directed  to  the  successive 
steps  taken  in  the  solution  of  the  problems  first  given  under  any  sub- 
ject, and  each  pupil  should  be  required  to  state  the  first  step,  the 
second  step,  and  so  on  until  all  the  steps  are  named  and  recorded  on 
slates  or  blackboard.  These  steps  should  be  combined,  and  thus  the 
mode  of  building  up  a  rule  be  made  clear.  The  rule  should  be  re- 
garded, not  as  a  guide  to  the  solution  of  problems,  but  as  a  concise 
statement  of  what  the  pupils  have  already  learned  to  do. 

5.  In  addition,  pupils  should  usually  avoid  naming  the  numbers  to 
be  added,  but  should  give  only  the  successive  results.  They  should 
have  much  practice  in  adding  and  subtracting  by  2's,  3's,  4's,  etc.  In 
multiplication,  they  may  profitably  use  either  form  of  expression — 
2  times  2,  3  times  2,  4  times  2,  etc. ;  or,  two  2's,  three  2's,  four  2's,  etc. 
Sometimes  one  of  these  expressions  is  preferable,  sometimes  the  other. 

6.  The  explanations  given  are  not  to  be  committed  to  memory.  The 
definitions  and  principles,  having  been  fully  comprehended,  ought  to  bb 
fixed  in  the  memory.  The  rules  may  or  may  not  be  learned,  as  teacliers 
shall  prefer. 

7.  Fractions  should  be  amply  illustrated  by  material  objects,  atten- 
tion being  specially  called  to  the  number  and  the  size  of  the  parts 
into  which  a  thing  is  divided. 

8.  Care  should  be  taken  that  the  explanations  given  by  pupils  are 
logical  in  order  and  accurate  in  expression. 

4 


CONTENTS. 


INTEGERS. 

SECTION  PA« 

I. — Numbers 7 

II. — Notation  and  Numeration 11 

III. — Addition 16 

IV.— Subtraction 27 

V.— Review 37 

VI. — Multiplication 40 

VII.— Division 52 

VIII.— Review 65 

FACTORING. 

IX.— Factors 68 

X.— Divisors 70 

XL — Multiples 72 

XII.— Cancellation 74 


FRACTIONS. 

XIII.— Notation 77 

XIV.— Reduction 82 

XV.— Addition 92 

XVI.— Subtraction 94 

XVII. — Multiplication 97 

XVIII.— Division 105 

XIX.— Review Ill 

UNITED  STATES  MONEY. 

XX.— Notation 116 

XXL— Reduction 119 

XXIL— Computations 121 

1*  5 


6  CONTENTS. 

DENOMINATE   NUMBERS. 

SECTION  PAG-i 

XXIII. — Measures  of  Extension 13< 

XXIV. — Measures  of  Capacity Y6V 

XXV. — Measures  of  Weight 142 

XXVI.— Circular  Measures „.  146 

XXVII. — Measures  of  Time 148 

XXVIII.— Paper  and  Counting 150 

XXIX.— Keview lt><> 


COMPOUND  NUMBERS. 

XXX.— Keduction 1j>4 

XXXI.— Addition 158 

XXXII.— Subtraction 161 

XXXI II.— Multiplication 165 

XXXIV.— Division 168 

XXXV.— Review 171 

DECIMALS. 

XXXVI. — Notation  and  Numeration 174 

XXXVII. — Addition  and  Subtraction 178 

XXXVIIL— Multiplication 180 

XXXIX.— Division 182 

XL. — Reduction 184 

XLI.— Review 187 

PERCENTAGE. 

XLII— Notation 188 

XLIII.— Cases 190 

XLIV—  Interest 1« 

XLV.— Review 19b 

APPENDIX. 

Rectangular  Measurements 201 

Miscellaneous  Problems 209 

Answers  to  Written  Exercises 215 


Elementary  Arithmetic. 


SECTION   I. 
NUMBERS. 

ARTICLE  1. — 1.  Arthur  has  one  white  rabbit  and 
one  gray  rabbit.     How  many  rabbits  has  he  ? 

2.  Arthur  has  two  rabbits,  and  his  sister  has 
one.     How  many  rabbits  have  both  of  them  ? 

3.  Jane  had  two  books,  and  her  father  gave  her  two 
more.     How  many  books  had  she  then  ? 

4.  If  you  have  three  cents  in  one  pocket,  and  two 
cents  in  another,  how  many  have  you  in  both  ? 

5.  James  gave  four  cents  for  an  orange,  and  two  cents 
for  an  apple.     How  many  cents  did  he  give  for  both  ? 


8  NUMBERS. 

6.  If  you  count  the  fingers  and  thumb  on  your  right 
hand,  how  many  will  you  find  them  to  be  ? 

7.  How  many  are  six  books  and  two  books  ? 

8.  Arthur  has  five  lead  pencils  and  four  slate  pencils. 
How  many  pencils  has  he  ? 

9.  Six  trees  and  three  trees  are  how  many  trees  ? 

10.  How  many  are  one  and  one?  Two  and  one? 
Three  and  one  ?     Four  and  one  ? 

11.  How  many  are  five  and  one?  Six  and  one? 
Seven  and  one?     Eight  and  one?     Nine  and  one? 

12.  How  many  are  two  and  two?  Three  and  two? 
Four  and  two  ?     Five  and  two  ?     Six  and  two  ? 

13.  How  many  are  seven  and  two?  Eight  and 
two  ?  Three  and  two  ?  Three  and  three  ?  Four  and 
three? 

14.  How  many  are  five  and  three  ?  Six  and  three  ? 
Seven  and  three  ?     Four  and  four  ? 

15.  How  many  are  five  and  four?  Six  and  four? 
Five  and  five  ? 

16.  Count  from  one  to  ten.  Count  the  fingers  and 
thumbs  which  you  have  on  both  hands. 

17.  If  you  had  five  books,  and  should  have  three 
more  given  you,  how  many  books  would  you  then 
have  ? 

18.  How  many  are  five  and  three  ?     Three  and  five  ? 

19.  How  many  are  six  and  two?  Eight  and  two? 
Four  and  three  ?     Seven  and  three  ? 

20.  If  you  have  six  books,  and  have  four  more  given 
you,  how  many  books  will  you  then  have  ? 


NUMBERS.  9 

2. — 1.  How  many  ones  in  a  collection  of  one  one  and 

one  one  ?     Of  two  ones  and  one  one  ? 

2.  How  many  ones  in  a  collection  of  nine  ones  and 
one  one  ?     Of  ten  ones  and  one  one  ? 

3.  How  many  are  nine  and  two  ?     Eight  and  three  ? 
Nine  and  three  ?     Seven  and  four  ? 

4.  How  many  dollars  are  six  dollars  and  four  dollars? 
Seven  dollars  and  three  dollars  ? 

5.  How  many  are   eight  and  four?     Six  and  five? 

Six  and  six  ? 

6.  How  many  are  eight   and   six?     Nine  and  six? 
Seven  and  seven?     Eight  and  seven? 

7.  How  many  are  eight  and  eight?     Nine  and  eight? 
Nine  and  nine  ? 

8.  How  many  are   ten   and   five?     Ten  and   four? 
Seven  and  four  ?     Nine  and  three  ? 

9.  How  many  are   five   and  five?     Nine  and  five? 
Seven  and  six  ?     Eight  and  six  ? 

10.  How  many  are  seven  and  seven  ?     Ten  and  six  ? 
Eight  and  eight? 

11.  How  many  are  six  and   ten?     Nine   and    six? 
Seven  and  eight  ?     Eight  and  nine  ? 

12.  How  many  are  eight  and  ten?     Ten  and  nine? 
Nine  and  ten  ?   Ten  and  ten  ?   Count  from  one  to  twenty. 

13.  How  many  tens  are  one  ten  and  one  ten  ?     Two 
tens  and  three  tens  ?     Count  from  one  to  thirty. 

14.  How  many  tens  are  four  tens  and   three  tens  ? 
Six  tens  and  four  tens  ?   Count  from  one  to  one  hundred. 


1U  NUMBERS. 

DEFINITIONS. 

3.  A  Unit  is  one,  or  a  single  thing  of  any  kind. 

4.  A  Number  is  a  unit,  or  a  collection  of  units. 

5.  A  Figure  is  a  character  used  to  express  a  number. 
Each  of  the  first  nine  numbers  is  expressed  by  u 

single  figure,  thus — 

printed.        1,      2,      3,       4,       5,      6,      7,       8,     9. 

WR.TTEN.         /234$6/8? 
NAMED.  One,       two,      three,      four,      five,        six,       seven,      eight,    nine. 

The  figure  0,  which  is  called  Zero,  or  Cipher,  ex- 
presses the  absence  of  number. 

1  ten  is  named      Ten.         6  tens  are  named  Sixty. 

2  tens  are  named  Twenty.    7  tens  are  named  Seventy. 

3  tens  are  named  Thirty.     8  tens  are  named  Eighty. 

4  tens  are  named  Forty.       9  tens  are  named  Ninety. 

5  tens  are  named  Fifty.     10  tens  are  named  One  hundred. 

When  a  number  is  expressed  by  two  figures,  side  by 
side,  the  figure  on  the  right,  or  in  the  first  place,  ex- 
presses Ones,  and  the  figure  on  the  left,  or  in  the  second 
place,  expresses  Tens. 

Thus,  57  expresses  5  tens  7  ones,  or  fifty-seven. 

Wit  I TTEX  EXJE  It  CIS  ES. 

6.  Copy  and  name  the  number  expressed — 


(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

19 

17 

16 

62 

75 

93 

91 

70 

74 

53 

42 

37 

23 

62 

57 

77 

57 

82 

36 

91 

80 

66 

90 

28 

41 

82 

11 

46 

28 

69 

55 

73 

85 

64 

82 

99 

NUMERATION  AND  NOTATION,  11 

SECTION   IX 

NUMERATION  AND  NOTATION. 

7. — 1.  By  combining  what  two  figures  do  we  express 
ten,  or  1  ten  .0  ones  ?     Eleven,  or  1  ten  1  one  ? 

2.  By  combining  what  two  figures  do  we  express 
twelve,  or  1  ten  2  ones  ?     Thirteen,  or  1  ten  3  ones  ? 

3.  By  combining  what  two  figures  do  we  express 
nineteen,  or  1  ten  9  ones?  Ninety-one,  or  9  tens  1 
one? 

4.  By  combining  what  two  figures  do  we  express 
seventy-five,  or  7  tens  5  ones  ?  Fifty-seven,  or  5  tens 
7  ones? 

5.  By  combining  what  two  figures  do  we  express 
eighty,  or  8  tens  0  ones  ?  Ninety-three,  or  9  tens  3 
ones? 

DEFINITIONS. 

8.  Numeration  is  the  method  of  naming  numbers. 

9.  Notation  is  the  method  of  writing  numbers. 

10.  In  Naming  Numbers,  ten  ones  are  named  one  ten,  ten 
tens  are  named  one  hundred,  and  so  on. 

11.  A  figure  written  alone,  or  at  the  left  of  a  point  (,\ 
called  the  Decimal  Point,  expresses  ones,  or  Primary  Units. 

12.  Orders  of  Units  are  expressed  by  the  successive 
figures  written  side  by  side  to  express  number. 

When  a  number  is  expressed  by  three  figures,  the 
first  figure  at  the  left  of  the  decimal  point  expresses 
units  of  the  First  Order,  or  Ones;  the  second  figure  at  the 
left  of  the  point  expresses  units  of  the  Second  Order,  or 


12  NUMERATION  AND  NOTATION. 

Tens;  the  third  figure  at  the  left  of  the  point  expresses 
units  of  the  Third  Order,  or  Hundreds,  and  so  on„ 

Thus,  365  expresses  5  units  of  the  first  order,  6  units  of  the 
second  order,  3  units  of  the  third  order,  or  three  hundred  sixty- 
five. 

13.  In  Reading  Numbers,  each  of  the  three  orders  of 
figures  at  the  left  of  the  decimal  point  expresses  a  Class 
or  Period  of  units,  with  a  distinct  name,  having  ones, 
tens  and  hundreds,  as  shown  in  the  following 

NUMEEATION  TABLE. 

PERIODS.  Millions.  Thousands.  Units. 


2  2  2  « 

ORDERS.  <j         rf         rf  I         «         *  •««».§ 

W^o  He<o  WehoA 

number.       35     6,       7     8     9,       403., 

where  the  number  expressed  by  the  figures  is  three 
hundred  fifty-six  millions  seven  hundred  eighty-nine 
thousands  four  hundred  three. 

14.  The  Comma  (,)  is  used  to  separate  the  classes  or 
periods,  and  the  Decimal  Point  (.)  to  mark  the  order  of 
primary  units. 

When  the  decimal  point  is  not  expressed  it  is  always 
understood. 
Thus,  32.  or  32  expresses  thirty-two. 

15.  A  Solution  is  the  process  of  answering  a  question 
requiring  computation. 

16.  A  Problem  is  a  question  for  solution. 

17.  A  Proof  of  a  solution  is  the  process  of  testing  its 
correctness. 


NUMERATION  AND  NOTATION 


13 


18.  A  Rule  is  a  concise  statement  of  the  method  of 
solving  a  problem. 

19.  A  Principle  is  a  general  or  settled  truth. 

Principle  of  Numeration  and  Notation. 

20.  Ten  units  of  any  order  are  equal  to  one  unit  of  the 
next  higher  order. 

WRITTEN  EXERCISES. 

21.  Copy  and  read — 


1.  100. 

8.  200. 

15.  323. 

22.   982. 

2.    101. 

9.  202. 

16.   175. 

23.   600. 

s.  116. 

10.  220. 

17.  555. 

24.   802. 

4.   126. 

11.   118. 

18.  400. 

25.    111. 

5.   161. 

12.  221. 

19.   ^0^. 

26.  787. 

6.   162. 

13.  212. 

20.  444* 

27.    P£l. 

7.   170. 

14.  300. 

21.  507. 

28.    i£0. 

22.  Write  i 

n  figures — 

1.  Sixty-two 

f 

r.  One  hundred  three. 

2.  Seventy-fi 

ve. 

i 

},  Two  hundred  fifteen. 

3.  Ninety. 

( 

).  Three  hundred  sixty. 

4.  Fifty-five 

1( 

).  Two  hundred  twenty-one. 

5.  Eighty. 

11 

..  Four  hundred  forty. 

6.  Eighty-eij 

rht. 

1: 

1.  Five  hundr 

gd  ninety. 

23.— 1.  Copy  and  read  316405. 

Solution.  —  Separating  the  given  figures  into  classes  or 
periods  of  three  figures  each,  we  have  316,405.  The  first  period 
at  the  left  of  the  decimal  point  expresses  405  units,  and  the 
second  period  316  thousands.  The  whole  is  read :  three  hundred 
sixteen  thousand  four  hundred  five. 
2 


14 


NUMERATION  AND   NOTATION. 


Copy  and  read — 

2.  314670. 

8. 

101452. 

14. 

99000 

3.  G5809. 

9. 

39000. 

15. 

131311 

4.  132735. 

10. 

7888. 

16. 

46006 

5.   3257. 

11. 

50005. 

17. 

800000. 

6.  625407. 

12. 

683452. 

18. 

7314. 

7.  700314. 

13. 

700000. 

19. 

660066. 

24. — 1.  Write  in  figures  three  hundred  sixteen  thou- 
sand four  hundred  five. 

Solution.— Writing  316  for  the  hundreds, 
®ip   /fir       tens  and  ones  of  thousands,   and  405  for  the 
hundreds,  tens  and  ones  of  units,  gives  316,405, 
the  required  expression. 

Write  in  figures — 

2.  Three  hundred  seventy-three.     Four  hundred. 

3.  Seven  hundred  sixty-one.     Three  hundred  one. 

4.  Eight  hundred  eighty-five.     Seventy-nine. 

5.  One  thousand  one  hundred  forty-nine. 

6.  Nine  thousand  three  hundred  thirty-three. 

7.  Twenty-five  thousand  three  hundred  sixty-five. 

25.  Rule  for  Numeration.— Beginning  with  the  lowest 
order  of  units,  separate  the  figures  of  the  given 
number  into  periods  of  three  figures  eaeh. 

In  reading,  begin  with  the  highest  period;  read 
the  hundreds,  tens  and  ones  of  eaeh  period ;  and 
give  the  name  of  eaeh  period,  except  units,  after 
its  ones. 

26.  Rule  for  Notation.—  Write  the  figures  expressing 
the  hundreds,  tens  and  ones  of  eaeh  period  in 
their  order. 


NUMERATION  AND   NOTATION 


15 


fltOBLMMS. 

27.  Write  and  read — 

1.      711. 

5.  11,224 

9.  100,000. 

2.      309. 

6.  12,560. 

10.  525,125. 

3.  1,624. 

7.  50,06*0. 

11.  500,025. 

4.  3,052. 

8.  60,606. 

12.  751,157. 

13.  3,163,000. 

14.  4,725,555. 

15.  65,008,721. 

16.  364,230,000. 
28.  Express  by  figures — 

1.  Five  hundred.    Five  thousand.    Five  hundred  five. 

2.  Five  thousand  five.     Eight  hundred  nine. 

3.  One  thousand  one  hundred  nineteen. 

4.  Two  thousands  five  hundreds  ninety. 

5.  Eight  hundreds  seventy-seven. 

6.  Twenty-five  thousands  eight  hundred  eight. 

7.  Fifty-nine  thousands  nine  hundreds. 

8.  One  hundred  sixty-three  thousands. 


29.  Test  Questions. — 1.  What  is  a  unit?  A  number? 
What  is  a  figure  ? 

2.  Name  the  first  nine  numbers.  What  numbers  are  ex- 
pressed by  a  single  figure?  When  a  number  is  expressed  by 
two  figures,  side  by  side,  what  does  the  figure  on  the  right  ex- 
press?    The  figure  on  the  left? 

3.  What  is  numeration?  Notation?  What  does  a  figure 
written  alone,  or  at  the  left  of  the  decimal  point,  express? 

4.  How  are  orders  of  units  expressed  ?  What  does  the  first 
figure  at  the  left  of  the  decimal  point  express?  The  second? 
The  third  ?     Recite  the  orders  of  the  Numeration  Table. 

5.  What  is  used  to  separate  figures  into  classes  or  periods1? 
What  is  used  to  mark  the  order  of  primary  units  ? 

6.  What  is  a  solution?  A  problem  ?  A  proof?  A  rule?  A 
principle? 


7.  What  is  a  principle  of  notation^ 


16  ADDITION. 

-      SECTION   III. 
ADDITION. 

30. — 1.  Jane  has  5  books,  and  her  sister  has  6.  How 
many  have  both  ? 

2.  Albert  has  7  white  roses  and  8  red  ones.  How 
many  roses  has  he  in  all  ? 

3.  How  many  are  5  and  6?  How  many  ones  are  8 
and  7? 

4.  How  many  cents  are  8  cents  and  4  cents  ?  How 
many  cents  are  4  cents  and  8  cents  ? 

5.  In  a  garden  there  are  10  pear  trees  and  7  peach 
trees.     How  many  trees  are  there  of  the  two  kinds  ? 

6.  A  man  bought  a  plow  for  10  dollars,  and  sold  it 
for  2  dollars  more  than  he  gave  for  it.  How  many  dol- 
lars did  he  get  for  it? 

7.  A  boy  rode  8  miles,  and  walked  7.  How  far  did 
he  travel  ? 

8.  In  a  certain  class  there  are  9  boys  and  8  girls. 
How  many  pupils  are  there  in  the  class  ? 

9.  How  many  ones  are  10  and  2  ?  10  and  7  ?  9 
and  8  ?     10  and  8  ? 

10.  How  many  ones  are  9  and  1  ?  10  and  2  ?  7  and 
4?     9  and  7?     10  and  6  ? 

11.  What  number  is  formed  by  uniting  the  ones  in 

6  and  5?     10  and  9? 

12.  What  number  is  formed  by  uniting  the  ones  in 

7  and  3?     10  and  3  ? 


ADDITION, 


17 


TABLES. 


0  and 

1  and 

2  and 

3  and 

4  and 

1  are   1 

1  are   2 

1  are   3 

1  are   4 

1  are   5 

2    "     2 

2    "     3 

2    "     4 

2    "     5 

2    "     6 

3    "     3 

3    "     4 

3    "     5 

3    "     6 

3    "     7 

4    «     4 

4    "     5 

4    "     6 

4    "    7 

4    "     8 

5    "     5 

5    "     6 

5    "     7 

5    "     8 

5    "     9 

6    "     6 

6    "     7 

6    "     8 

6    "     9 

6    "  10 

7    "     7 

7    "     8 

7    "     9 

7    "  10 

7    "  11 

8    "    8 

8    "     9 

8    "  10 

8    "  11 

8    "  12 

9    "     9 

9    "  10 

9    "  11 

9    "  12 

9    "  13 

10    "  10 

10    "  11 

10    "  12 

10    "  13 

10    "  14 

5  and 

6  and 

7  and 

8  and 

9  and 

1  are   6 

1  are   7 

1  are   8 

1  are   9 

1  are  10 

2    "     7 

2    "     8 

2    "     9 

2    "  10 

2    "  11 

3    "     8 

3    "     9 

3    "  10 

3    "  11 

3    "  12 

4    "     9 

4    "  10 

4    "  11 

4    "  12 

4    "  13 

5    "  10 

5    "  11 

5    "  12 

5    "  13 

5    "  14 

6    "  11 

6    "  12 

6    "  13 

6    "  14 

6    "  15 

7    "  12 

7    "  13 

7    "  14 

7    "  15 

7    «  16 

8    "  13 

8    "  14 

8    «  15 

8    "  16 

8    "  17 

9    "14 

9    "  15 

9    "  16 

9    "  17 

9    "  18 

10    "  15 

10    "  16 

10    "  17 

10    "  18 

10    "  19 

DEFINITIONS. 

31.  The  Sum  of  two  or  more  numbers  is  a  number 
containing  as  many  units  or  ones  as  those  numbers. 

32.  Addition  is  the  process  of  uniting  two  or  more 
numbers  to  find  their  sum. 

33.  The  Sign  of  Addition  is  an  upright  cross,  -{-,  and  is 
called  plus. 

34.  The  Sign  of  Equality  is  =,  and  is  read  equals  or  equal 

Thus,  9  -f-  5  =  14,  is  read  nine  plus  five  equals  fourteen,  and 
means  that  5  added  to  9  equals  14. 


2* 


18  ADDITION. 


MENTAL    EXERCISES. 


35. — 1.  If  Henry  has  9  dollars  and  his  brother  has  7, 
how  many  dollars  have  both  ? 

Solution. — If  Henry  has  9  dollars  and  his  brother  has  7, 
both  have  as  many  dollars  as  the  sum  of  9  and  7,  which  is  16, 
the  answer  required. 

2.  If  you  gather  7  quarts  of  berries  and  your  brother 
6  quarts,  how  many  do  both  of  you  gather  ? 

3.  A  boy  caught  7  fishes,  and  another  boy  caught  5. 
How  many  did  both  catch  ? 

4.  Harry  had  8  cents,  and  had  5  more  given  him. 
How  many  had  he  then  ? 

5.  How  many  are  11  and  3  ?     5  and  8  ?     10  and  5  ? 

6.  John  has  6  cents,  George  5  and  Victor  2.  How 
many  cents  have  they  in  all  ? 

7.  If  I  give  to  one  boy  5  peaches,  to  another  7,  and 
keep  4,  how  many  peaches  had  I  at  first  ? 

8.  How  many  are  5,  7  and  4  ?  7,  5  and  4  ?  5,  4 
and  7?     7,  4  and  5?     4,  5  and  7?     4,  7  and  6? 

9.  How  many  are  9  and  4?  6,  3  and  2?  3,  6 
and  4? 

10.  How  many  are  13  and  2  ?    15  and  2  ?    15  and  4  ? 

11.  What  does  11  +  7  mean?     8  +  3  =  11  ? 

12.  A  boy  paid  5  cents  for  an  orange,  7  cents  for  a 
melon  and  6  cents  for  a  pencil.  How  many  cents  did 
he  pay  for  them  all  ? 

13.  How  many  are  16  and  2  ?  16  and  4?  17  and 
2?     17  and  3?     18  and  2? 


ADDITION.  19 


WRITTEN  EXERCISES. 


36.  Copy  and  add — 

(1.) 
1 

~         Solution. — Begin  at  the  bottom  of  the  column,  and 

3      say,  zero,  three,  five,  six. 

0         Write  6  under  the  line  at  the  bottom  of  the  column. 

6 

(2.)  (3.)  (4.)  (5.)  (6.)  (7.)  (8.) 

5  4-5314-7 

7  3  3  2  3  2  0 

2  2  2  8  9  0  7 

117  6  7  9  5 


15 

17 

20 

19 

37.  Copy  and  add — 

(1.)           (2.) 

(3.) 

(4.) 

.(5.) 

(6.) 

(7.) 

8           5 

9 

4 

6 

2 

9 

2           3 

0 

1 

6 

1 

0 

6           0 

7 

2 

2 

8 

9 

4           9 

3 

4 

2  ■ 

7 

0 

20  19  16  18 

Add  each  of  the  above  columns  downward.  If  the 
work  is  correct,  the  result  will  be  the  same  as  that  first 
obtained. 

MENTAL    EXERCISES. 

38.— 1.  How  many  are  10  and  5  ?     20  and  5  ? 

2.  How  many  are  30  +  5?     12  +  6  ?     11  +  7  ? 

3.  Count  by  2's  from  2  to  12. 
Solution. — Two,  four,  six,  eight,  ten,  twelve. 


20  ADDITION. 

4.  Count  by  2's  from  12  to  30.     From  30  to  50. 

5.  Count  by  4's  from  4  to  40.     From  40  to  80. 

6.  Count  by  5?s  from  25  to  55.     From  30  to  60. 

7.  Count  by  6's  from  2  to  32.     From  30  to  60. 

8.  How  many  are  9  and  2  ?  19  and  2  ?  29  and  2  ? 
9  and  3?     19  and  3?     39  and  3? 

9.  How  many  are  9  and  4  ?  29  and  4  ?  49  and  4  ? 
9  and  5  ?  39  and  5  ?  59  and  5  ?  19  and  5  ?  39  and 
7?     69  and  7? 

10.  How  many  are  9  and  6  ?  49  and  6  ?  79  and  6  ? 
9  and  8  ?  19  and  8  ?  29  and  8  ?  9  and  9  ?  29  and 
9  ?     89  and  7  ? 

11.  On  one  branch  of  a  tree  are  16  apples,  on  another 
6,  and  on  a  third  2.     How  many  apples  are  there  in  all  ? 

39.— 1.  Count  by  7's  from  3  to  45.     From  70  to  98. 

2.  Count  by  8's  from  1  to  41.     From  64  to  96. 

3.  How  many  are  S  and  6  ?  48  and  6  ?  78  and  6  ? 
8  and  7?     38  and  7?     58  and  7?     8  and  8?    48  and  8? 

4.  How  many  are  8  and  9  ?  28  and  9  ?  68  and  9  ? 
8  and  10?     18  and  10?     78  and  10? 

5.  If  a  barrel  of  flour  is  worth  12  dollars,  a  hundred- 
weight of  fish  5  dollars  and  a  cheese  4  dollars,  how 
many  dollars  are  all  worth  ? 

6.  What  are  such  numbers  as  express  things  of  the 
same  kind,  as  6  cents  and  3  cents,  or  7  tens  and  4  tens, 
called  ?  Similar  Numbers. 

7.  Can  you  add  6  cents  and  7  tens  ?  Can  you  add  6 
cents  and  7  cents  ? 

8.  What  is  the  sum  of  4  +  3  +  2?  Of  3  + 2  +  4? 
Of  2  + 3  + 4?     Of  3  +  4+ 2?     Of  4+ 2 +  3? 


ADDITION.  21 

9.  How  may  the  correctness  of  an  answer  in  addition 
be  tested  ? 

By  adding  the  numbers  a  second  time  and  in  a  different  order. 

10.  Add  3,  4  and  11,  and  prove  the  correctness  of 
the  answer. 

11.  A  man  gave  some  apples  to  2  boys;  to  one  he 
gave  4,  and  to  the  other  24.  If  he  should  give  9  more 
to  each,  how  many  would  each  have  ? 

Principles  of  Addition. 

40. — 1.   Only  similar  numbers  can  be  added. 
2.   The  sum  of  numbers  is  the  same,  in  whatever  order 
they  are  added. 

WRITTEN  EXERCISES. 

41.  Copy,  add  and  prove — 

ID  (2.)  (3.)  (4.)  (5.)  (6.) 


s 

4 

5 

3 

2 

9 

3 

2 

7 

0 

7 

3 

4 

1 

2 

5 

3 

4 

1 

3 

0 

7 

4 

5 

6 

7 

9 

8 

4 

7 

42.  Write,  add  and  prove — 

(l.) 

16  Solution. — Write  the  numbers  so  that  the  figures  ex- 
®2  pressing  ones  stand  in  one  column,  and  the  figures  ex- 
Qp.      pressing  tens  stand  in  another  column,  at  the  left. 

.  Begin  with  the  ones,  and  add  the  ones  and  tens  separ- 

138      ately. 
Write  the  sum  of  the  ones  under  the  line  at  the  bottom  of  the 

column  of  ones. 


ADDITION. 


Write  the  sum  of  the  tens  under  the  line  at  the  bottom  of  the 
column  of  tens. 
The  result  is  138,  which  is  the  sum  required. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

51 

40 

44 

13 

47 

91 

62 

51 

31 

32 

72 

15 

34 

73 

JJ 

51 

50 

12 

147 

118 

(8.) 

(9.) 

(io:) 

(11.) 

(12.) 

(13.) 

23 

U 

42 

60 

44 

17 

70 

31 

85 

70 

11 

61 

96 

43 

10 

19 

33 

50 

189 
43.  Write  in  columns,  add  and  prove — 
43,  91,  60  and  U. 
50,  75,   11  and  32. 
42,  12,  13  and  80. 

4.  81,  17,  41  and  70. 

5.  55+71  +  32  +  20. 

6.  63  +  80  +  60+  17. 


128 


7.  16  +  81  +  12  +  50. 

8.  60+73+15  +  31. 

9.  43  +  60+72+  19. 

10.  50+    7  +  91  +  10. 

11.  92+11  +  33+  12. 

12.  80+15+12+  71. 


MEXTAI,    EXERCISES. 

44. — 1.  James  paid  45  cents  for  a  reading-book  and 
17  cents  for  a  writing-book.  How  many  cents  did  he 
pay  for  both  ? 

Solution. — He  paid  as  many  cents  as  the  sum  of  45  and  17. 
45  and  10  are  55 ;  55  and  7  are  62 :  therefore  he  paid  62  cents. 

2.  If  you  should  ride  23  miles  and  walk  13  miles, 
how  far  would  you  travel  ? 


ADDITION.  23 

3.  How  many  are  27  +  24?     CI  +  9?     61  +  29? 

4.  George  had  30  chickens ;  he  bought  20  more,  and 
had  1 1  more  given  him.     How  many  had  he  then  ? 

5.  How  many  are  7  and  9  ?     17  and  9  ?     27  and  9  V 

6.  How  many  are  16  and  5  ?     36  and  5  ?    6  and  6  t 

7.  I  bought  some  beef  for  60  cents,  some  veal  for  30 
cents,  and  a  fish  for  15  cents.  How  much  did  the  whole 
cost? 

8.  Count  by  9's  from  9  to  63.     From  63  to  108. 

9.  I  paid  80  dollars  for  a  cow  and  25  dollars  for  a 
heifer.     How  much  did  both  cost  ? 

10.  How  many  are  6  and  7  ?    16  and  7  ?    36  and  7  ? 

11.  How  many  are  6  and  9  ?  26  and  9  ?  86  and  9? 
6  and  10?     36  and  10  ?     76  and  10  ? 

WRITTEN   EXERCISES. 

45.— 1.  Add  855,  345  and  64. 

{855         Solution.— Write  the  numbers  so  that 
345      %ures  expressing  units  of  the  same  order 
&  ,      stand  in  the  samo  column. 
Begin  with  the  ones,  and  add  the  ones, 
Sum,      1264-      tens  and  hundreds  separately. 
The  sum  of  the  ones  is  14  ones,  or  1  ten  4  ones. 
Write  the  4  ones  under  the  line  at  the  bottom  of  the  column 
of  ones,  and  add  the  1  ten  with  the  tens  of  the  next  column. 
The  sum  of  the  tens  is  16  tens,  or  1  hundred  6  tens. 
Write  the^6  tens  under  the  line  at  the  bottom  of  the  column 
of  tens,  and  add  the  1  hundred  with  the  hundreds  of  the  next 
column. 
The  sum  of  the  hundreds  is  12  hundreds. 
Write  the  12  hundreds  under  the  line  at  the  bottom  of  the 
column  of  hundreds. 

The  result  is  1264,  which  is  the  sum  required. 


24 


ADDITION. 

i  and 

explain  in 

like  mannei 

132 

13.) 
186 

14.) 

144 

(5.) 
191 

(6.) 

443 

46' 

304 

555 

49 

37 

53 

80 

342 

723 

8 

46.  Rule  for  Addition.—  Write  the  numbers  so  that  all 
the  figures  of  the  same  order  shall  stand  in  the 
same  column,  and  draw  a  line  under  the  columns. 

Begin  at  the  right,  add  the  units  of  each  order 
separately,  and  write  their  sum,  if  less  than  ten, 
under  the  column  added. 

If  the  sum  of  the  units  of  any  order  be  ten  or 
more,  write  the  figure  standing  for  its  ones,  and, 
add  its  tens  with  the  units  of  the  next  higher  order. 

Writ.e  the  whole  sum  of  the  units  of  the  highest 
order. 

Proof. — Add  the  numbers  a  second  time,  in  a  different  order. 
The  result  should  be  the  same  by  both  methods. 

PROBLEMS. 

47.  Find  the  sum — 


4.  Of  27,  366  and  5549. 

5.  Of  340,  3331  and  19: 

6.  Of  633,  902  and  1187. 


1.  Of  615,  3045  and  5000. 

2.  Of  309,  446  and  7131. 

3.  Of  3241,  445  and  199. 

48.  Add,  explain  and  prove — 

(1.)  (2.)  (3.)  (4.) 

134  dollars.        472  men.        160  miles.        341  bushels. 
452      "  306     "  49     "  428      " 

317      "  28     "  672     "  152      " 


ADDITION,'. 

25 

(5.) 

173  feet 

(6.) 
3142  tons. 

(7.) 
9162  bales. 

(8.) 
149  horses. 

416    " 

4152     " 

713      " 

934      " 

712    " 

613    " 

44     " 

196      " 

49.— 1.  What  is  the  sum  of  9163  +  120  +  315  +  4  ? 

2.  What  is  the  sum  of  13162  +  417  +  491  +  82? 

3.  Add  1345,  7819,  100,000,  and  316,271. 

(4)                           (5.)  (6.)                      (7.) 

145  yards.  1000  pounds.  2183  men.  315  cords. 

315      "  5006       "  19     "  705     " 

410      "  3508       "  _94     "  814     " 

8.  What  is  the  sum  of  9168  +  429  +  251  +  128  ? 

9.  What  is  the  sum  of  3162  +  414  +  490  +  18  ? 

10.  Add  fifteen  thousand  three  hundred  forty-two  and 
nine  thousand  nine  hundred  sixty-six. 

11.  Add  one  million  one  thousand  nine,  five  hundred 
thousand  seven  hundred  eleven  and  ninety-five  thousand 
four  hundred  four. 

50. — 1.  Washington  was  born  in  the  year  1732,  and 
lived  67  years.     In  what  year  did  he  die  ? 

2.  A  farmer  paid  5750  dollars  for  his  farm  and  375 
dollars  for  tools.     What  was  the  cost  of  the  whole  ? 

3.  In  one  school  are  516  pupils,  in  another  314,  and 
in  another  215.     How  many  pupils  are  there  in  all  ? 

4.  One  steamer  has  on  board  1321  bales  of  cotton,  a . 
second  3150  bales  and  a  third  5725  bales.     How  many 

ales  have  the  three  on  board  ? 

5.  Smith  has  1631-  bushels  of  wheat,  Jones  1500 
shels,  Woodman  783  bushels,  and  Kaspar  1735 
shels.     How  many  bushels  have  they  in  all  ? 


26  ADDITION. 


6.  If  }Tou  should  pay  175  dollars  for  a  horse,  225 
dollars  for  a  yoke  of  oxen,  80  dollars  for  a  cart  and  93 
dollars  for  a  wagon,  how  much  would  they  all  cost  you? 

51.  Write,  add  and  prove — 


(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

12 

31 

145 

123 

1412 

3 

92 

54 

356 

3400 

17 

16 

63 

720 

570 

90 

42 

20 

114 

145 

43 

15 

40 

333 

7682 

7 

72 

68 

121 

1031 

82 

13 

37 

301 

915 

(6.)  (7.)  (8.)  (9.)  (10.) 

25  42  191  333  1614 

19  52  37  404  1641 

8  71  170  500  901 


U 

55 

45 

608 

109 

62 

83 

68 

171 

3170 

20 

17 

22 

349 

143 

9 

71 

11 

100 

28 

52.  Test  Questions. — 1.  What  is  the  sum  of  two  or  more 
numbers  ?    What  is  addition  ? 

2.  What  is  the  sign  of  addition?  What  is  it  called?  What 
is  the  sign  of  equality?     How  is  it  read? 

3.  What  are  such  numbers  as  express  things  of  the  same  kind 
called?  How  may  the  correctness  of  an  answer  in  addition  be 
tested? 

4.  What  principles  of  addition  are  given  ? 


SUBTRACTION. 


27 


SECTION    IV. 
SUBTRACTION. 


53. — 1.  John  had  2  ap- 
ples, and  gave  1  of  them 
to  his  sister.  How  many 
had  he  left? 

2.  A  boy  had  3  apples,  and  sold  2  of  them.      How 
many  had  he  left? 

3.  James  had  4  marbles,  and  lost  2  of  them.     How 
many  had  he  left? 

4.  John  has  6  rabbits,  and  his  sister  has  3.     How 
many  more  has  John  than  his  sister  ? 

6.  If  I  sold  what  cost  me  5  dollars  for  7  dollars,  how 
many  dollars  more  than  the  cost  did  I  receive  ? 

6.  If  you  have  8  acorns,  and  should  give  5  to  your 
brother,  how  many  would  you  have  left  ? 

7.  A  hen  had  8  chickens,  but  a  hawk  took  4  of  them. 
How  many  chickens  were  left  ? 


28 


SUBTRACTION. 


8.  A  boy  is  6  years  old  ;  in  how  many  years  will  he 
be  10? 

9.  William  is   9  years   old,   and   his  brother  is  11. 
What  is  the  difference  in  their  ages  ? 

10.  If  you  have  5  dollars,  and  should  lose  1  dollar, 
how  many  dollars  would  you  have  left. 

11.  A  merchant  bought  goods  for  9  dollars.     How 
much  will  he  gain  by  selling  them  for  11  dollars  ? 

12.  How  many  are  8  less  4  ?     9  less  7  ?     11  less  9  ? 

TABLES. 


1  from 

2  from 

3  from          4  from          5  from 

1  leaves  0 

2  leaves  0 

3  leaves  01  4  I 

eaves  0    5  leaves  0 

2      "      1 

3      "      1 

4     "      1 

5 

a 

1 

6      "      1 

3      "      2 

4     "     2 

5      "      2 

6 

it 

2 

7      "     2 

4      "      3 

5     "     3 

6      "      3 

7 

it 

3 

8     "     3 

5      "     4 

6      "     4 

7      "      4 

8 

it 

4 

9      "     4 

6      "     5 

7     "      5 

8      "      5 

9 

tt 

5 

10     "     5 

7      "     6 

8     "      6 

9      "      6 

10 

a 

6 

11      "     6 

8     "      7 

9     "      7 

10     "      7 

11 

a 

7 

12     "     7 

9     "     8 

10     "     8 

11      "      8 

12 

it 

8 

13     "     8 

10     "     9 

11      "      9 

12      "      9 

13 

it 

9 

14     "     9 

11      "   10 

12     "    10 

13     "    10 

14 

it 

10 

15     "    10 

6  from 

7  from 

8  from 

9  from 

10  from 

6  leaves  0 

7  leaves  0 

8  leaves  0 

9  leaves  0 

10  leaves  0 

7      "     1 

8      "      1 

9      "      1 

10 

a 

1 

11      "      1 

8      "     2 

9      "      2 

10      "      2 

11 

a 

2 

12     "     2 

9     "     3 

10     "     3 

11      "     3 

12 

it 

3 

13     "     3 

10     "     411      "     4 

12     "      4 

13 

a 

4 

14     "     4 

11      "     512     "     5 

13     "      5 

14 

n 

5 

15    "      5 

12     "     6  13     "     6 

14     "     6 

15 

n 

6 

16     "      6 

13     "     7 

14     "      7 

15     "     7 

16 

a 

7 

17     "     7 

14     "     8 

15      "      8 

16      "     8 

17 

it 

8 

18      "      8 

15     "     9 

16      "     9 

17      "      9 

18 

it 

9 

19      "      9 

16     "    10 

17      "    10 

18      "    10 

19 

ti 

10 

20     "    10 

SUBTRACTION.  29 

DEFINITIONS. 

54.  The  Difference  between  two  numbers  is  the  num- 
ber which,  when  added  to  the  less,  will  make  the 
greater. 

55.  Subtraction  is  the  process  of  finding  the  difference 
between  two  numbers. 

56.  The  Minuend  is  the  number  from  which  the  sub- 
traction is  made. 

57.  The  Subtrahend  is  the  number  which  is  subtracted. 

58.  The  Sign  of  Subtraction  is  — ,  and  is  called  minus. 
When  placed  between  two  numbers  it  denotes  that  the 
one  on  the  right  of  it  is  to  be  taken  from  that  on  the  left. 

Thus,  13  —  5  is  read,  thirteen  minus  five,  and  means  that  5  is 
to  be  subtracted  from  13. 

MENTAL   EXERCISES. 

59. — 1.  If  a  boy  had  16  dollars  and  lost  9  of  them, 
how  many  had  he  left  ? 

Solution. — If  a  boy  had  16  dollars,  and  lost  9  of  them,  he 
had  left  as  many  as  the  difference  between  16  and  9,  which  is  7, 
the  answer  required. 

2.  I  bought  a  book  for  20  cents  and  sold  it  for  12 
cents.     How  many  cents  did  I  lose  by  the  sale  ? 

•  3.  William  has  14  dollars  in  one  purse  and  24  dollars 
in  another.  If  he  should  take  5  dollars  from  each,  how 
many  dollars  would  be  left  in  them  ? 

4.  A  farmer  having  17  dollars,  paid  10  dollars  for 
flour.  How  many  dollars  had  he  left?  10  and  how 
many  are  17? 

5.  T  paid   19  dollar?  for  a  coat  and  7  dollars  for  a 

3  * 


30  SUBTRACTION. 

vest.     How  many  dollars  more  were  paid  for  the  coat 
than  for  the  vest  ?     7  and  how  many  are  19  ? 

Principles  of  Subtraction. 

60. — 1.  Only  similar  numbers  can  be  subtracted,  the 
one  from  the  other. 

2.  The  difference  and  the  subtrahend  are  together  equal 
to  the  minuend. 

WRITTEN    EXERCISES. 

61.  Write  and  subtract — 

(1.)  Solution. — Write  the  numbers  for  sub- 

From  -i9      trading  so  that  the  figures  representing  ones 

rp  ,  qr      stand  in  one  column,  and  the  figures  repre- 

senting  tens  stand  in  another  column  at  the 

Difference,      23      left. 

Begin  with  the  ones,  and  subtract  the  ones  and  the  tens 
separately. 

Write  the  difference  of  the  ones,  which  is  3,  under  the  line  at 
the  bottom  of  the  column  of  ones. 

Write  the  difference  of  the  tens,  which  is  2,  under  the  line  at ' 
the  bottom  of  the  column  of  tens. 

The  result  is  23,  which  is  the  difference  required. 

(2.)  (3.)  (4.)  (5.)  (6) 

From  97  82  63  66  97 

Take  43  51  43  23  W 

Difference,      54  20  27 

(7.)  (8.)  (9.)  (10.) 

From  48  books.  72  yds.      55  lbs.  96  dollars. 

Take  13     "  61    "         44    "  16_      " 

Add  the  subtrahend  and  difference  in  each  of  the  last 
four  problems.  Their  sum  should  be  equal  to  the 
minuend. 


SUBTRACTION.  31 

MENTAL   EXERCISES. 

62.— 1.  How  many  are  12  less  5?     14  less  8?     18 
less  8? 

2.  How  many  are  10  less  five?      10  less  8/      13 
less  9? 

3.  Count  backward  by  2's,  or  by  subtracting  2  suc- 
cessively, from  22  to  2. 

Solution.— 22,  20,  18, 16,  14, 12, 10,  8,  6,  4,  2. 

4.  Count  backward  by  3's  from  60  to  30.     From  30 
to  3. 

5.  Count  backward  by  4's  from  48  to  8.     From  96 
to  72. 

6.  Count  backward  by  5's  from  40  to  10.     From  80 
to  55. 

7.  Count  backward  by  6's  from  90  to  12.     From  35 
toll. 

8.  Count  backward  by  7's  from  61  to  5.     From  70 
to  35. 

9.  Count  backward  by  9's  from  45  to  9.     From  72 
to  45. 

10.  How  many  will  remain  if  2  be  taken  from  11  ? 

2  from  21  ?     2  from  31  ?     2  from  41  ? 

11.  How  many  will  remain  if  3  be  taken  from  11  ? 

3  from  21  ?     3  from  41  ?     3  from  61  ? 

12.  How  many  will  remain  if  4  be  taken  from  11  ? 

4  from  21  ?     4  from  31  ?     4  from  51  ? 

63. — 1.     What  number  must  be  added  to  7  to  make 
13  ?     To  make  23  ?     To  make  33  ? 

2.  What  number  must  be  added  to  6  to  make  14  ? 


32  SUBTRACTION 

3.  If  the  subtrahend  is  9  and  the  difference  8,  what 
is  the  minuend  ? 

4.  What  number  is  19  —  8?     17  —  9?     16  —  7? 

5.  What  number  must  be  added  to  8  to  make  13  ? 
To  make  23  ?     To  make  33  ? 

6.  How  many  are  21  less  9  ?     13  less  9  ? 

7.  What  number  must  be  added  to  9  to  make  13? 
To  make  23  ? 

8.  If  you  have  9  dollars,  how  many  more  must  you 
have  to  make  16  dollars?     To  make  26  dollars? 

9.  If  you  are  now  8  years  old,  in  how  many  years 
will  you  be  14  ?     In  how  many  will  you  be  24  ? 

10.  Jacob  has  6  lambs;  how  many  more  must  he 
have  to  make  12  ?     To  make  22  ? 

11.  2  tens  from  4  tens  leave  how  many  tens  ?  2  tens 
from  6  tens  ?    5  tens  from  9  tens  ?    7  tens  from  1 1  tens  ? 

12.  A  shepherd  has  20  sheep  in  one  fold,  and  30  in 
another.  How  many  more  has  he  in  the  second  than 
in  the  first  ? 

13.  John  has  50  cents ;  how  many  more  must  he 
have  to  make  70  cents  ? 

WRITTEN  EXERCISES. 

64. — 1.  What  is  the  difference  between  86  and  47  ? 

Solution. — Write  the  figures  of  the  sub- 
7  16      trahend  under  the  figures  of  the  same  order 
Minuend,        86      in  the  minuend. 

Subtrahend,  47  Since  7  units   cannot  be  taken   from   6 
units,  take  1  ten  from  the  8  tens  of  the  minu- 

Difference,      39      en^  leaving  7  tens,  and  add  10  ones,  which 

Proof  86      are  e<lual  to  the  1  ten,  to  the  6  ones,  thus 

making  the  minuend  the  same  as  7  tens,  16 


SUBTRACTION.  33 

Subtract  7  ones  from  16  ones,  leaving  9  ones,  which  write 
under  the  line  at  the  bottom  of  the  ones. 

Subtract  4  tens  from  7  tens,  leaving  3  tens,  which  write  under 
the  line  at  the  bottom  of  the  tens. 

The  result  is  39,  which  is  the  difference  required. 

Prove  the  solution  by  adding  the  difference  to  the  subtrahend, 
which  gives  86,  the  minuend. 

Subtract,  explain  in  like  manner  and  prove — 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

Minuend, 

93 

56 

74 

87 

260 

Subtrahend, 

18 

29 

65 

78 

49 

Difference, 

75 

9 

211 

(V.) 

(8.) 

(9.) 

(10.) 

(ll.) 

Minuend, 

363 

457 

554 

934 

918 

Subtrahend, 

254 

148 

535 

828 

909 

Difference, 

109 

19 

9 

MENTAL   EXERCISES. 

65. — 1.  A  farmer  had  25  bushels  of  wheat,  and  sold 
12  bushels.     How  many  bushels  had  he  left  ? 

Solution. — He  had  left  as  many  bushels  as  the  difference 
between  25  and  12;  12  =  10  +  2;  10  from  25  leaves  15,  and  2 
from  15  leaves  13,  which  is  the  answer  required. 

2.  If  you  had  29  dollars,  how  many  dollars  would 
you  have  left  after  spending  16  dollars? 

3.  What  number  must  be  added  to  21  to  make  40  ? 

4.  What  number  must  be  added  to  13  to  make  61  ? 

5.  What  number  must  be  added  to  49  to  make  76  ? 

6.  Arthur  had  28  cents,  and  his  father  gave  him 
enough  more  to  make  50.  How  many  cents  did  his 
father  give  him? 


34  SUBTRACTION. 

7.  I  had  39  sheep ;  having  sold  some,  I  found  there 
were  only  23  left.     How  many  had  been  sold  ? 

8.  Mary's  mother  is  41  years  old;  Mary  is  13  years 
aid.  In  how  many  years  will  Mary  be  as  old  as  her 
mother  now  is  ? 

9o  From  a  cask  containing  63  gallons,  25  gallons 
leaked  out.     How  many  gallons  remained  in  the  cask  ? 

10.  If  you  had  47  dollars,  and  should  give  away  15, 
how  many  dollars  would  you  have  remaining  ? 

11.  A  man  sold  31  cattle  from  a  drove  of  80.  How 
many  remained  ? 

12.  In  an  orchard  containing  44  trees,  25  are  apple 
trees,  and  the  rest  are  pear  trees.  How  many  are  pear 
trees? 

13.  I  bought  a  horse  for  90  dollars,  and  sold  him  at 
a  loss  of  37  dollars.  For  how  many  dollars  was  the 
horse  sold  ? 

14.  The  cost  of  a  carriage  was  100  dollars  and  that 
of  a  harness  43  dollars.  How  many  dollars  did  the 
carriage  cost  more  than  the  harness  ? 

15.  A  drover  bought  some  cattle  for  100  dollars,  and 
sold  them  for  88  dollars.  How  many  dollars  did  he 
lose? 

16.  If  you  should  exchange  a  watch  worth  75  dollars 
for  a  carriage  worth  93  dollars,  how  many  dollars  would 
you  make  by  the  trade  ? 

17.  How  much  is  made  by  selling  a  horse  which  cost 
93  dollars,  for  105  dollars  ? 

18.  I  bought  a  lot  of  land  for  115  dollars,  and  sold  it 
for  85  dollars.    How  much  did  I  lose  by  the  transaction  ? 


SUBTRACTION.  35 

WRITTEN  EXERCISES. 

66.— 1.  From  4334  subtract  2530. 

Solution. — 0  ones  from  4  ones  leaves 

3-  *3*  4  ones ;  3  tens  from  3  tens  leaves  0  tens ; 

Minuend,        4334      5  hundreds  cannot  be  subtracted  from  3 

Subtrahend,  2530      hundreds ;  therefore  take  1  thousand  from 

Difference,  1804  the  4  thousands  of  the  minuend,  leaving 
3  thousands,  and  add  10  hundreds,  which 
equal  1  thousand,  to  the  3  hundreds,  thus  making  the  minu- 
end the  same  as  3  thousands  13  hundreds  3  tens  4  ones ;  5  hun- 
dreds from  13  hundreds  leaves  8  hundreds,  and  2  thousands  from 
3  thousands  leaves  1  thousand. 
The  result  is  1804,  which  is  the  difference  required. 

Subtract  and  explain  in  like  manner — 


2.  4562  from  6278. 

3.  835  from  5734. 

4.  2071  from  7665. 


6.  1205  from  8304. 

7.  600,  from  9502. 

8.  1835  from  7035. 


5.   3473  from  3743.      1      9.   801  from  8010. 

67.  Rule  for  Subtraction.—  Write  the  subtrahend  under 
the  minuend,  so  that  the  figures  of  the  same  order 
shall  stand  in  the  same  column. 

Begin  at  the  right,  and  subtract  the  units  of  each 
order  of  the  subtrahend  from  the  units  of  the  same 
order  in  the  minuend,  if  possible,  and  write  the 
difference  under  the  order  subtracted. 

When  the  units  of  any  order  of  the  minuend  are 
less  than  those  of  the  same  order  of  the  subtrahend, 
increase  their  number  by  adding  ten,  the  value  of  a 
unit  taken  from  the  next  higher  order  of  the  minu- 
end, and  subtract  ;  then  consider  the  units  of  that 
higher  order  of  the  minuend  one  less. 


36  SUBTRACTION. 

Proof. — Add  the  difference  and  subtrahend  together.    Theii 
sum  should  be  equal  to  the  minuend. 

PROBLEMS. 

68.  Find  the  difference  between — 


1.  1863  and  945. 

2.  1345  and  807. 

3.  87203  and  68315. 


4.  81992  and  85. 

5.  6447  una  5446. 

6.  35702  and  16205. 


69.  Perform  the  subtraction  indicated,  and  explain — 


1.  43402  —  1233. 

2.  7206  —  1506. 

3.  8200  —  3415. 

4.  6302  —  4444. 


5.  10931  —  7301. 

6.  14275  —  3900. 

7.  4466  —  2271. 

8.  90043  —  73132. 


70. — 1.  A  man  born  in  the  year  1809  died  in  the  year 
1870.     How  many  years  did  he  live? 

2.  A  farmer  bought  a  farm  for  6390  dollars,  and  sold 
it  for  9500  dollars.  How  much  did  he  gain  by  the 
transaction  ? 

3.  From  a  cargo  of  corn  containing  1563  bushels, 
752  bushels  have  been  taken.  How  many  bushels 
remain  ? 

4.  A  certain  mountain  is  18635  feet  high,  and  another 
mountain  is  15367  feet  high.  What  is  the  difference  in 
their  heights  ? 

5.  A  merchant  sold  for  10560  dollars,  goods  which 
cost  him  8775  dollars.  How  much  did  he  gain  by  the 
transaction  ? 


REVIEW.  37 

6.  How  many  is  1000  —  90  ?     1000  —  11? 

7.  From  nineteen  thousands  take  nineteen. 

8.  A  city  has  149306  inhabitants,  which  is  21319 
more  than  it  had  five  years  ago.  How  many  inhabitants 
had  it  then  ? 


71.  Test  Questions.— 1.  What  is  the  difference  between  two 
numbers  ?  What  is  subtraction  ?  What  is  the  minuend  ?  the 
subtrahend  ? 

2.  What  is  the  sign  of  subtraction  ?  What  principles  of  sub- 
traction are  given  ? 


SECTION   V. 
REVIEW. 
72. — 1.  By  what  process  do  you  find  the  sum  of  two 
or  more  numbers  of  the  same  kind  ? 

2.  By  what  process  do  you  find  the  difference  between 
two  numbers  of  the  same  kind  ? 

3.  How  do  Addition  and  Subtraction  differ  ? 

4.  Can  you  add,  or  unite  into  one  number,  numbers 
of  different  kinds,  as  4  feet  and  5  dollars  ? 

5.  Can  you  take  one  number  from  another  of  a  differ- 
ent kind,  as  4  feet  from  5  dollars  ? 

6.  A  man  bought  a  barrel  of  flour  for  17  dollars, 
some  butter  for  9  dollars,  some  molasses  for  6  dollars 
and  some  sugar  for  5  dollars.  How  many  dollars  did 
the  whole  cost  him  ? 

7.  In  a  yard  are  26  hens,  6  turkeys  and  4  ducks. 
How  many  fowls  are  in  the  yard  ?  How  many  would 
there  be  left  if  a  hawk  should  take  7  of  them  ? 

4 


38  REVIEW 

8.  I  bought  28  cows  at  one  time,  7  at  another  and  6 
at  another,  and  afterward  sold  9.     How  many  remained  ? 

9.  Frank  had  45  cents;  he  spent  11  cente  for  a  writ- 
ing-book, 8  cents  for  a  pencil  and  5  cents  for  a  balL 
How  many  cents  had  he  left  ? 

10.  There  are  24  hours  in  a  day;  if  you  should 
spend  10  hours  in  sleep,  3  in  work  and  5  in  play,  how 
many  hours  will  you  have  left  for  study  ? 

11.  A  company  went  into  battle  with  93  men ;  20  of 
them  were  killed  and  10  were  taken  prisoners.  How 
many  were  left  ? 

12.  George  had  50  cents  to  spend;  he  bought  a  ball 
for  10  cents,  a  pen  for  5  cents,  a  pencil  for  3  cents  and 
a  piece  of  rubber  for  8  cents.  How  many  cents  had  he 
left? 

13.  If  from  27  you  take  15  less  8,  how  many  will 
remain  ? 

14.  A  boy  bought  some  articles  for  48  cents ;  for  how 
many  cents  must  he  sell  them  to  gain  12  cents  ?  For 
how  many  cents  to  lose  9  cents  ? 

15.  Alfred  lost  5  cents  and  found  15  cents,  and  then 
had  40  cents.     How  many  cents  had  he  at  first  ? 

16.  What  number  must  be  taken  from  39  to  leave  10 
more  than  25  ? 

17.  What  number  must  be  added  to  31  to  make  a 
sum  11  less  than  51  ? 

18.  Susan  and  Mary  have  each  17  books;  if  Susan 
should  give  5  of  hers  to  Mary,  how  many  more  would 
Mary  have  than  Susan  ? 


REVIEW.  39 

WRITTEN  EXERCISES. 

73. — 1.  What  is  the  sum  of  three  hundred  sixty-five, 
one  thousand  six  hundred  fifty,  and  three  hundred 
twenty-five  ? 

2.  If  you  have  1600  dollars,  and  should  gain  1734 
dollars,  how  much  money  would  you  then  have  ? 

3.  If  you  have  3334  dollars,  and  should  lose  2500, 
how  much  would  you  have  left  ? 

4.  The  minuend  is  11567  and  the  subtrahend  7457, 
What  is  the  difference  ? 

5.  The  subtrahend  is  7457  and  the  difference  4110, 
What  is  the  minuend  ? 

6.  The  difference  is  4110  and  the  minuend  is  11567. 
What  is  the  subtrahend  ? 

7.  Reed  had  1367  tons  of  coal;  he  sold  at  one  time 
325  tons  and  at  another  572  tons.  How  many  tons  had 
he  then  left? 

Solution. — He  sold  at  one 

time  325  tons,  and  at  another 

572  tons.  1367  tons.       572  tons  J  hence  he  sold  325  tons 

QOg    "  897     u         "^  ^^  ^ons>  which  are  897  tons. 

Since  he  had  1367  tons  and  sold 

897  tons.  470  tons.       897  tons,  he  then  had  left  the 

difference   between    1367    tons 
and  897  tons,  which  is  470  tons. 

8.  I  had  3675  dollars,  and  gained  1320  dollars.  If 
I  should  pay  away  2720  dollars,  how  much  would  I 
have  left? 

9.  How  much  is  3675  —  539  added  to  363  +  73  ? 

10.  How  much  is  the  sum  of  7867  and  1319,  dimin- 
ished by  4261 ? 


40 


MULTIPLICATION. 


SECTION   VI. 

MUL  TIPLICA  TION. 

74. — 1.  If  John  can  remove  2  books  at  one  time,  how 
many  can  he  remove  at  two  times  ? 

2.  How  many  are  2  books  and  2  books,  or  2  times  2 
books  ? 

3.  If  you  have  2  cherries,  and  James  has  3  times  as 
many,  how  many  has  James  ? 

4.  How  many  are  2  and  2  and  2,  or  3  times  2  ? 

5.  If  you  have  4  fingers  on  each  hand,  how  many 
have  you  on  both  hands  ? 

6.  John  has  3  blocks,  and  his  brother  has  3  times  as 
many.     How  many  has  his  brother  ? 

7.  If  you  should  get  3  merit-marks  each  day,  how 
many  would  you  get  in  4  days  ? 

8.  How  many  are  4  times  3  ?     3  times  4  ? 


M  UL  Tl PLICA  TIOX. 


41 


9.  On  one  twig  there  are  4  cherries  ;    how   many 
cherries  are  there  on  2  similar  twigs  ? 

10.  How  many  cents  are  3  times  4  cents  ? 

11.  There  are  four  pecks  in  one  bushel ;  how  many 
pecks  are  there  in  4  bushels  ? 

12.  When  6  cents  are  paid  for  1  quart  of  milk,  how 
much  must  be  paid  for  2  quarts  ? 

13.  How  many  are  2  times  6  ?     How  many  cents  are 

2  times  6  cents  ? 

14.  How  many  are  5  and  5  and  5,  or  3  times  5  ? 

15.  At  the  rate  of  5  dollars  a  week,  how  many  dollars 
can  be  earned  in  3  weeks  ? 

75. — 1.  How  many  are  once  1?     Once  2?     Once  3? 
Once  4? 

2.  How  many  are  2  times  1  ?     2  times  2  ?     2  t* 
3?     2  times  4?     2  times  5?  l  8 

3.  How  many  are  2  times  6  ?     2  times  7  ?     2  ti 
8?     2  times  9?     2  times  10? 

4.  How  many  are  3  times  1  ?     3  times  2  ?     3  times 
3?     3  times  4?     3  times  5? 

5.  How  many  are  3  times  6  ?     3  times  7  ?     3  times 
8  ?     3  times  9  ?     3  times  10  ? 

6.  How  many  are  4  times  1  ?     4  times  2?     4  times 

3  ?     4  times  4  ?     4  times  5  ? 

7.  How  many  are  4  times  6  ?     4  times  7  ?     4  times 
8?     4  rimes  9?     4  times  10? 

8.  How  many  are  5  times  1  ? 
3  ?     5  times  4  ?     5  times  5  ? 

9.  How  many  are  5  times  6  ? 
8?     5  times  9?     5  times  10? 

4  » 


5  times  2  ?     5  times 
5  times  7  ?     5  times 


42 


MUL  TIPLICA  TION. 


10.  How  many  are  2  times  2  ?     3  times  2  ?     4  times 
2  ?    ,5  times  2  ? 

11.  How  many  are  6  times  2  ?     7  times  2  ?     8  times 
2?"    9  times  2?     10  times  2? 

12.  How  many  are  2  times  3  ?     3  times  3  ?     4  times 
3?     5  times  3? 

13.  How  many  are  6  times  3  ?     7  times  3  ?     8  times 
3?     9  times  3?     10  times  3? 

TABLES. 


Once 

2  times 

3  times 

4  times 

5  times 

6  times 

1    is    1 

1  are   2 

1  are   3 

1  are  4 

1  are   5 

1  are   6 

2   "    2 

2  " 

4 

2  "     6 

2  "     8 

2  "   10 

2  "  12 

3   "    3 

3  " 

6 

3  "     9 

3  "  12 

3  "   15 

3  "   18 

4   "    4 

4  " 

8 

4  "   12 

4  "  16 

4  "  20 

4  "  24 

5   "    5 

5  " 

10 

5  "  15 

5  "  20 

5  "  25 

5  "  30 

"    6 

6  " 

12 

6  "  18 

6  "  24 

6  "  30 

6  "  36 

«    7 

7  " 

14 

7  "  21 

7  "  28 

7  "  35 

7  "  42 

8 

8  " 

16 

8  "  24 

8  "  32 

8  "  40 

8  "  48 

'    9 

9  " 

18 

9  "  27 

9  "  36 

9  "  45 

9  "  54 

74  «  io 

10  " 

20 

10  "  30 

10  "  40 

10  "  5o 

10  "  60 

111!       "11 

11  " 

22 

11  "  33 

11  "  44 

11  "  55 

11  "  66 

i2   "  12 

12  " 

24 

12  "  36 

12  "  48 

12  "  60 

12  "  72 

7  times 

8  times 

9  times 

10  times 

11  times 

12  times 

lare  7 

lare  8 

lare  9 

1  are  10 

lare  11 

lare  12 

2  "  14 

2  " 

16 

2  "  18 

2  "   20 

2  "   22 

2  "    24 

3  "  21 

3  " 

24 

3  "  27 

3  "   30 

3  "   33 

3  "   36 

4  "  28 

4  " 

32 

4  "  36 

4  "   40 

4  "   44 

4  "    48 

5  "  35 

5  " 

40 

5  "  45 

5  "   50 

5  "   55 

5  "    60 

6  "  42 

6  " 

48 

6  "  54 

6  "   60 

6"    66 

6  "    72 

7  "  49 

7  " 

56 

7  «  63 

7  "   70 

7  "    77 

7  "   84 

8  "  56 

8  " 

64 

8  "  72 

8  "   80 

8  "   88 

8  "   96 

9  "  63 

9  " 

72 

9  "  81 

9  "   90 

9  "   99 

9  "  108 

10  "  70 

10  " 

80 

10  "  90 

10  "  100 

10  "110 

10  "  120 

11  "  77 

M  " 

88 

11  "  99 

11  "110 

11  "121 

11  "132 

12  "  84 

12  " 

96 

12  "108 

12  "120 

12  "132 

12  "144 

MUL  TIPLICA  TION.  43 

DEFINITIONS. 

76.  Multiplication  is  the  process  of  taking  one  of  t^o 
numbers  as  many  times  as  there  are  units  in  the  other. 

77.  The  Multiplicand  is  the  number  to  be  multiplied. 

78.  The  Multiplier  is  the  number  by  which  to  multiply. 

79.  The  Product  is  the  result  of  the  multiplication. 
The  multiplicand  and  multiplier  are  Factors  of  the  product. 

80.  The  Sijrn  of  Multiplication  is  an  oblique  cross,  X, 
mid  is  *ead  times  or  multiplied  by. 

Thus,  9  X  8  is  read  nine  multiplied  by  eight,  or  eight  times  nine. 

MENTAL,    EXERCISES. 

81.— 1.  What  is  the  product  of  9  by  7  ?    Of  5  by  4  ? 

2.  What  is  the  product  of  7  by  5  ?     Of  6  by  3  ? 

3.  If  1  orange  cost  6  cents,  how  many  cents  will  8 
oranges  cost  ?  * 

Solution. — If  1  orange  cost  6  cents,  8  oranges  will  cost  8 
times  6  cents,  which  are  48  cents. 

4.  If  you  can  earn  4  dollars  in  1  week,  how  many 
dollars  can  you  earn  in  6  weeks  ? 

5.  What  will  8  pairs  of  boots  cost  at  7  dollars  a  pair  ? 

6.  How  many  trees  in  9  rows  of  6  trees  each  ? 

7.  How  many  yards  are  7  times  10  yards? 

8.  At  4  dollars  each,  what  will  11  barrels  of  apples 
cost? 

9.  James  had  12  chickens,  and  his  brother  6  times  as 
many.     How  many  chickens  had  his  brother  ? 

10.  At  7  cents  each,  what  will  9  oranges  cost  ?  * 

11.  At  9  cents  each,  what  will  9  pencils  cost  ? 


44  M  UL  TIP LIC A  TION. 

12.  If  a  horse  eat  6  quarts  of  oats  in  1   day,  how 
many  quarts  will  he  eat  in  6  days  ? 

13.  At  10  cents  each,  what  will  8  pencils  cost  ? 

14.  If  11  yards  of  calico  be  required  for  1  dress,  how 
many  yards  will  be  required  for  3  dresses  ? 

15.  If  12  cents  must  be  paid  for  a  quart  of  cherries, 
how  many  cents  must  be  paid  for  9  quarts  ? 

WRITTEN  EXERCISES. 

82.  Copy  and  multiply — 
(1.) 
Multiply      6         Solution.— Three  times  six  are  eighteen. 
By  o  Write  18  under  the  line  at  the  bottom  of  the 

Product,    18      column' 

(2.)  (3.)  (4.) 

Multiply      5                   7  8 

By    '         _4  j*  S 

Product,    20  64 

(7.) 
Multiply    23         Begin  with  the  ones,  and  multiply  the  ones 
gy  g      and  the  tens  separately. 

Write  the  product  of  the  ones  by  3,  which  is 
Product,    69      9?  Under  the  column  of  ones. 

Write  the  product  of  the  tens  by  3,  which  is  6,  under  the 
column  of  tens. 

The  result  is  69,  which  is  the  product  required. 


(5.) 

(6. 

9 

6 

6 

5 

30 

(8.) 

(9.) 

(10.) 

(11.) 

(12.) 

(18. 

Multiply   4.3 

21 

31 

41 

50 

60 

E,    '          2 

A 

6 

5 

7 

8 

Product,  86 

186 

350 

MUL  TIP LIC A  TION.  45 


MENTJLL    EXERCISES. 

83. — 1.  What  is  the  product  of  9  and  8  multiplied 
together  ? 

2.  What  is  the  product  of  3X5?     Of  3X5X6? 

3.  What  is  the  product  of  4  X  3  X  2  ?     Of  3  X  4 
X2?     Of  2X3X4?     Of  3X2X4? 

4.  Is  the  product  of  the  factors  4,  3  and  2  the  same 
if  multiplied  together  in  different  orders  ? 

5.  In  a  certain  garden  there  are  12  rows,  with  11  trees 
in  each  row.     How  many  trees  are  there  in  the  garden  ? 

6.  James  is  12  years  old,  and  his  uncle  is  5  times  as 
old.     What  is  the  age  of  his  uncle  ? 

7.  If  you  earn  10  dollars  in  1  month,  how  much  can 
you  earn  in  10  months  ? 

8.  How  many  cents  must  be  paid  for  12  yards  of 
muslin  at  12  cents  a  yard  ? 

9.  How  much  must  be  paid  for  9  quarts  of  cherries 
at  11  cents  a  quart  ? 

Principle  of  Multiplication. 

84.  The  product  of  factors  is  the  same  in  whatever  order 
they  are  multiplied. 

WRITTEN  EXERCISES. 

85.— 1.  Multiply  87  by  4. 

Solution.— The  product  of  the  7  ones 

Multiplicand,      87      b?  4  is  28  ones>  or  2  tens  8  ones* 

m»  7/.-  ?:  ~  /  Write  8  under  the  line  at  the  bottom 

Multiplier,  A        m  ,  „  ,  ..     ft 

of  the  column  of  ones,  and  reserve  the  2 

Product,  348      tens  to  unite  with  the  tens  of  the  next 

product. 


46 


MULTIPLICA  TION. 


The  product  of  the  8  tens  by  4  is  32  tens,  which,  with  the  2 
tens  that  were  reserved,  make  34  tens. 

Write  34  under  the  line  at  the  bottom  of  the  column  of 
tens. 

The  result  is  348,  which  is  the  product  required. 

Write  and  solve  in  like  manner — 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

Multiply 

93 

68 

75 

34 

92 

By 

5 

6 

8 

7 

9 

Product,  465 

600 

828 

(7.) 

(8.) 

(9.) 

(10.) 

Multiply 

125  dollars. 

84  bushels.    1 19  barrels. 

137 'tons. 

By 

3 

7 

2 

5 

Product,  375  dollars. 

86.— 1.  Multiply  25  by  10.     By  30. 


685  tons. 


Multiplicand,  25 

Multiplier,  10 

Product,  250 

Multiplicand,  25 

Multiplier,  SO 

Product,  750 


Solution. — To  multiply  by  10,  simply 
annex  a  cipher  to  the  multiplicand,  be- 
cause the  annexing  of  a  cipher  removes 
each  figure  one  order  to  the  left,  and 
makes  the  value  denoted  tenfold  as  large 
as  before. 

Solution. — Since  30  is  3  X  10,  the 
product  of  25  by  30  must  be  the  product 
of  25  X  3  X  10.  3  times  25  is  75,  and  10 
times  75  is  750. 


Write  and  multiply  in  like  manner — 

5.     31  by  100. 


2.  125  by  10. 

3.  134hy40. 

4.  133  by  20. 


6.  62  by  500. 

7.  124  by  700. 


M  UL  TI PLICA  TION.  47 

MENTAL   EXERCISES. 

87. — 1.  If  it  take  6  men  8  days  to  do  a  piece  of 
work,  how  many  days  will  it  take  1  man  to  do  it  ? 

Solution. — It  will  take  1  man  6  times  as  long  as  6  men.  If 
it  take  6  men  8  days  to  do  a  piece  of  work,  it  will  take  1  man  6 
times  8  days,  which  are  48  days. 

2.  If  9  men  can  mow  a  field  in  3  days,  in  what  time 
can  1  man  mow  it  ? 

3.  How  many  days  must  1  man  work  to  do  a  job 
which  would  require  the  labor  of  7  men  for  5  days  ? 

4.  If  a  certain  amount  of  provisions  will  last  9  per- 
sons 10  days,  how  long  will  it  last  1  person  ? 

5.  George  is  8  years  old,  and  his  father  is  4  years 
more  than  3  times  as  old.  What  is  the  age  of  his 
father? 

6.  If  1  orange  be  worth  6  apples,  how  many  apples 
must  be  given  for  9  oranges  ? 

7.  If  you  earn  12  dollars  a  month,  and  spend  7 
dollars  a  month,  how  many  dollars  can  you  save  in  12 
months  ? 

8.  When  the  price  of  apples  is  3  cents  each,  and  of 
lemons  5  cents  each,  what  will  5  apples  and  6  lemons 
cost? 

9.  If  you  can  earn  40  cents  a  day,  and  you  pay  30 
cents  a  day  for  your  board,  how  many  cents  can  you 
save  in  6  days  ? 

10.  If  it  take  12  men  to  reap  a  field  in  6  days,  how 
many  men  will  it  take  to  reap  it  in  1  day  ? 

11.  I  bought  3  plows  at  12  dollars  each,  and  gave  in 
payment  4  ten-dollar  bills.  How  much  change  ought 
to  be  paid  back  to  me  ? 


48  MUL  TIP LIC A  TION. 

12.  I  bought  8  oranges  at  5  cents  each,  and  gave  in 
payment  a  fifty-cent  piece.  How  much  change  should 
be  paid  back  to  me  ? 

13.  How  many  dollars  must  be  paid  for  5  tons  of 
coal  at  6  dollars  a  ton,  and  4  cords  of  wood  at  3  dollars 
a  cord  ? 

14.  Mary  is  10  years  old,  and  her  mother  is  4  years 
less  than  4  times  as  old.    What  is  the  age  of  her  mother  ? 

15.  When  blueberries  are  9  cents  a  quart,  and  black- 
berries 12  cents,  how  much  more  will  8  quarts  of  black- 
berries cost  than  8  quarts  of  blueberries  ? 

WRITTEN    EXERCISES. 

88.— 1.  Multiply  364  by  42. 

Solution. — 42  is  equal  to  4  tens  and 
Multiplicand,  364  2  ones.  364  multiplied  by  42  must 
Multiplier,  42      equal  2  times  364  -f  4  tens  times  364. 

p      .  7       ~.   — V&q  Write  the  multiplier  under  the  mul- 

f         /~o      tiplicand,  placing  figures  of  the  same 
Products,    j   1456         or(ier  in  the  same  column. 
Product  15288  ^he  Pr°duct  of  364  by  the  2  ones  is 

728  ones,  which  is  a  partial  product. 

4  tens  times  4  ones  are  16  tens,  or  1  hundred  +  6  tens ;  write 
6  for  the  tens  of  a  second  partial  product,  and  reserve  the  1 
hundred  to  unite  with  the  product  of  the  hundreds;  4  tens 
times  6  tens  are  24  hundreds,  which,  with  1  hundred  reserved,  are 
25  hundreds,  or  2  thousands  +  5  hundreds  ;  write  5  for  the  hun- 
dreds of  the  second  partial  product,  and  reserve  the  2  thousands 
to  unite  with  the  product  of  the  thousands ;  4  tens  times  3  hun- 
dreds are  12  thousands,  which,  with  2  thousands,  are  14  thou- 
sands ;  write  14  for  the  thousands  of  the  second  partial  product, 
which  gives  1456  tens,  or  14560. 

Find  the  required  product  by  adding  the  two  partial  products, 
which  gives  15288.     Prove  the  Work  by  carefully  reviewing  it. 


MVL  TIPLICA  TION. 


49 


Multiply  and  prove  in  like  manner — 


2.  214  by  14. 

3.  509  by  29. 

4.  114  by  16. 


5.  232  by  ^i. 

6.  444  by  55. 

7.  420  by  17. 


8.  ^5  by  25,  and  prove  by  reversing  the  order  of 
the  factors. 


Multiplicand, 
Multiplier, 
Partial 
Products, 

Product, 


43 

25 


\    215 
s,  j     86 


25 
43 


215 
86 

1075 


Proof  A      75 

hoo 

\1075 


Solution. —  Reverse 
the  order  of  the  factors 
by  taking  25  for  the 
multiplicand  and  43  for 
the  multiplier. 

If  the  work  is  correct, 
the  result  must  be  the 


same  by  both  processes,  because  the  product  of  any  set  of  factors 
must  be  the  same  in  whatever  order  they  may  be  multiplied. 

9.  67  by  39,  and  prove  by  reversing  the  order  of 
the  factors. 

89.  Rule  for  Multiplication.  —  Write  the  multiplier 
under  the  multiplicand,  placing  figures  of  the 
same  order  in  the  same  column. 

If  the  multiplier  consists  of  but  one  order  of  units, 
multiply  each  order  of  the  multiplicand  in  succes- 
sion, beginning  at  the  right,  by  the  multiplier, 
writing  the  right-hand  figure  of  the  result  under 
the  order  multiplied,  and  uniting  the  units  ex- 
pressed by  the  left-hand  figure,  if  any,  with  the 
next  result. 

If  the  multiplier  consists  of  more  than  one  order 
of  units,  multiply  each  order  of  the  multiplicand 
by  each  order  of  the  multiplier,  write  the  right- 
hand  figure  of  each  partial  product  under  the 


50  M  Vh  TIPLICA  TION. 

order  of  the  multiplier,  and  add  the  partial  prod- 
ucts. 

If  either  factor  have  one  or  more  ciphers  on  the  right,  multi- 
ply without  regard  to  the  cipher  or  ciphers  on  the  right  of  either 
factor,  and  annex  to  the  result  as  many  ciphers  as  there  are  on 
the  right  of  the  factors. 

Proof. — Review  the  work  carefully,  or  reverse  the  order  of 
the  factors  and  multiply.  The  result  should  be  the  same  by 
both  methods. 

PKOBIjjEMS. 

90.  Find  the  product  of — 

1.  1251  dollars  by  6. 

2.  9 03  hogsheads  by  4> 

3.  1037  yards  by  7. 

4.  4009  bushels  by  9. 

91. — 1.  What  is  the  product  of  642  multiplied  by 
403? 
Multiplicand,         642 

^      '  -it Omit  to  multiply  by  the  0  tens,  be- 

1926     cause  the  product  of  any  number  by 
2568  0  is  0. 


5.  365  days  by  27. 

6.  58  miles  by  43. 

7.  731  tons  by  1000. 

8.  3140  acres  by  120. 


Product,       258726 

2.  What  is  the  product  of  316  multiplied  by  502  ? 

3.  What  is  the  product  of  1207  X  2001  ? 

4.  How  many  are  1005  times  1005? 

5.  What  is  the  product  of  56390  multiplied  by  401  ? 

6.  What  will  125  horses  cost  at  108  dollars  each? 

7.  How  many  pounds  of  beef  are  there  in  107  barrels, 
if  each  barrel  contains  200  pounds  ? 


M  UL  TIPLICA  TION.  51 

8.  How  many  bricks  in  203  loads,  if  each  load  con- 
tains 1037  bricks  ? 

92. — 1.  What  will  a  farm  of  164  acres  cost  at  25 
dollars  per  acre  ? 

2.  How  many  soldiers  in  16  regiments,  if  each  regi- 
ment contains  819  men  ? 

3.  If  sound  travel  1120  feet  in  1  second,  how  many 
feet  will  it  travel  in  105  seconds  ? 

4.  The  multiplier  is  68  and  the  multiplicand  is  4320. 
What  is  the  product  ? 

5.  What  is  the  product  of  one  hundred  thirteen  mul- 
tiplied by  itself? 

6.  How  many  cents  will  3184  bushels  of  corn  cost  at 
87  cents  a  bushel  ? 

7.  What  is  the  product  of  555  X  44  X  6  ? 

8.  In  a  certain  storehouse  there  are  21  apartments, 
and  in  each  apartment  15  bins,  each  bin  containing  490 
bushels  of  wheat.  How  many  bushels  of  wheat  are 
there  in  all  ? 

9.  A  merchant  shipped  11109  boxes  of  sugar,  each 
weighing  198  pounds.  What  was  the  weight  of  the 
whole  ? 

10.  If  light  travel  at  the  rate  of  192,000  miles  in  a 
second,  how  far  will  it  travel  in  5  minutes  of  60  seconds 
each  ? 

93.  Test  Questions. — 1.  What  is  multiplication?  The 
multiplicand?    The  multiplier?    The  product? 

2.  What  is  the  sign  of  multiplication?  How  is  it  read? 
How  does  it  differ  in  form  from  the  sign  of  addition  ? 

3.  What  are  the  factors  of  a  product?  Name  a  principle  of 
multiplication. 


DIVISION. 


SECTION    VII. 
DIVISION. 

94. — 1.  Two  cows  have  4  horns.    How  many  horns  has 
each  cow? 

2.  How  many  times  2  horns  are  4  horns  ? 

3.  A  farmer  has  9  sheep.     How  many  times  3  sheep 
has  he  ? 

4.  Nine  sheep  are  how  many  times  3  sheep  ? 

5.  A  harrow  has  12  teeth.     How  many  times  3  teeth 
has  it  ?     How  many  times  4  teeth  has  it  ? 

6.  Twelve    is    how   many   times    4?      How   many 
times  3  ? 

7.  Eight  fowls  are  how  many  times  2  fowTls  ? 

8.  A  house  has  16  windows.     How  many  times  4 
windows  has  it  ? 


DIVISION.  53 

95. — 1.  How  many  times  1  in  2?     In  3?     In  4? 
In  5  ?     In  6  ? 

2.  How  many  times  2  in  2  ?     In  4  ?     In  5  ?     In  6  s? 
In  8?     In  10? 

3.  How  many  times  2  in  12  ?     In  14  ?     In  16  ?     In 
18?     In  20? 

4.  How  many  times  3  in  3  ?     In  6  ?     In  9  ?     In  12  5 
In  15? 

5.  How  many  times  3  in  18  ?     In  21  ?     In  24  ?     In 
27?     In  30? 

6.  How  many  times  4  in  4?     In  8  ?     In  12?     In 
16?     In  20? 

7.  How  many  times  4  in  24  ?     In  28  ?     In  32  ?     In 
36?     In  40? 

8.  How  many  times  5  in  5?     In  10?     In  15?     In 
20?     In  25? 

9.  How  many  times  5  in  30  ?     In  35  ?     In  40  ?     In 
45?     In  50?     In  60? 

10.  What  is  one  of  the  2  equal  parts  of  6  ?     Of  10? 
Of  16? 

11.  What  is  one  of  the  3  equal  parts  of  12  ?     Of  15  ? 
Of  24? 

12.  What  is  one  of  the  4  equal  parts  of  8  ?     Of  16  ? 
Of  40? 

13.  What  is  one  of  the  5  equal  parts  of  10  ?     Of  20  ? 
Of  35  ?     Of  45  ? 

14.  What  is  one  of  the  6  equal  parts  of  6  ?     Of  12  ? 
Of  24?     Of  30? 

15.  What  is  one  of  the  7  equal  parts  of  7  ?     Of  21  ? 
Of  35?     Of  42? 

16.  What  is  one  of  the  8  equal  parts  of  8  ?     Of  32  ? 

5* 


54 


DIVISION. 


TABLES. 


lin 

2  in 

Sin 

4  in 

5  in 

1,     Once. 

2, 

Once. 

3,     Once. 

4, 

Once. 

5,    Once. 

2iy     Z  times. 

4, 

2  times. 

6,     2  times. 

8, 

2  times. 

10,    2  times. 

3,   3   " 

6, 

3    " 

9,   3   " 

12, 

3    " 

15,   3    " 

4,    4    " 

8, 

4    " 

12,   4   " 

16, 

4    " 

20,  4    " 

5,   5   " 

10, 

5    " 

15,   5   " 

20, 

5    " 

25,   5    " 

6,   6   " 

12, 

6    " 

18,   6    " 

24, 

6    " 

30,   6    " 

7,    7    " 

14, 

7    " 

21,   7   " 

28, 

7    " 

35,   7    " 

8,   8   " 

16, 

8    " 

24,   8   " 

32, 

8    " 

40,   8    " 

9,    9    " 

18, 

9    " 

27,    9   " 

36, 

9    " 

45,   9    " 

10,10   " 

20,10    " 

30,10   " 

40, 

10    " 

50,10    " 

6  in 

7  in 

8  In 

9  in 

10  in 

6,     Once. 

v, 

Once. 

8,     Once. 

9, 

Once. 

10,  Once. 

12,     2  times. 

14, 

2  times. 

16,     2  times. 

18, 

2  times. 

2i\Jj  Jtiines. 

18,  3   " 

21, 

3   " 

24,   3   " 

27, 

3   " 

30,3   " 

24,  4   " 

28, 

4   " 

32,   4   " 

36, 

4   " 

40,4   " 

30,  5   " 

35, 

5    " 

40,   5   " 

45, 

5   " 

50,  5   " 

36,   6   " 

42, 

6   " 

48,    6    " 

54, 

6   " 

60,  6   " 

42,   7   " 

49, 

7   " 

56,    7   " 

63, 

7   " 

70,  7   « 

48,   8    " 

56, 

8   " 

64,   8   " 

72, 

8    " 

80,  8   " 

54,   9   " 

63, 

9   " 

72,    9   " 

81, 

9   « 

90,  9   " 

60,10   " 

70, 

10   " 

80,10   " 

90, 

10   " 

100,10  " 

96. — 1.  If  any  number  of  things  are  separated  into 
two  equal  parts,  what  is  each  part  called  ? 

One  half  of  the  number  of  things. 

2.  If  Arthur  has  6  apples,  and  should  distribute  them 
equally  between  2  friends,  how  many  apples  would  each 
friend  receive? 

Solution. — If  Arthur  has  6  apples,  and  should  distribute 
them  equally  between  2  friends,  each  friend  would  receive  one 
half  of  6  apples,  which  is  3  apples. 

3.  If  you  should  wish  to  distribute  10  cents  equally 


DIVISION.  oa 

between  2  boys,  what  part  of  10  cents  would  each  boy 
receive  ? 

4.  If  you  have  12  pears,  and  should  give  one  half 
of  them  to  your  brother,  how  many  would  he  receive  ? 

5.  If  any  number  of  things  be  separated  into  three 
equal  parts,  what  is  each  part  called  ? 

One  third  of  the  number  of  things. 

6.  If  3  boys  should  share  12  apples,  equally  dividing 
them,  what  part  of  12  apples  would  each  have? 

7.  What  is  one  third  of  18  dollars  ?  Of  24  gallons  ? 
Of  30  bushels  ?     Of  36  days  ? 

8.  When  any  number  is  separated  into  four  equal 
parts,  what  is  each  part  called  ? 

One  fourth  of  that  number. 

9.  If  20  pears  be  shared  equally  among  4  boys,  what 
part  of  the  20  pears  will  each  boy  have  ? 

10.  How  many  is  one  fourth  of  16  ?     Of  24  ? 

11.  If  any  number  be  separated  into  five  equal  parts, 
what  is  each  of  the  parts  called  ? 

One  fifth  of  the  number. 

12.  How  many  is  one  fifth  of  35  ?     Of  45  ?     Of  50  ? 

13.  When  any  number  is  separated  into  six  equal 
parts,  into  seven  equal  parts,  etc.,  what  is  each  of  these 
parts  called  ? 

One  sixth,  one  seventh,  etc.,  of  the  number. 

14.  What  is  one  sixth  of  48  ?  Of  60  ?  One  seventh 
of  21?    Of  35?     One  eighth  of  24?     Of  40?     Of  72? 

15.  If  6  hats  cost  48  dollars,  what  will  1  hat  cost? 

16.  What  is  one  tenth  of  30  ?     Of  100  ?     Of  120  ? 

17.  If  10  barrels  of  flour  cost  100  dollars,  what  will 
1  barrel  cost  ? 


56  DIVISION. 

DEFINITIONS. 

97. — 1.  Division  is  the  process  of  finding  how  many 
times  one  of  two  given  numbers  is  contained  in  the 
other;  or, 

Division  is  the  process  of  separating  one  of  two  given 
numbers  into  as  many  equal  parts  as  there  are  units  in 
the  other. 

98.  The  Dividend  is  the  number  to  be  divided. 

99.  The  Divisor  is  the  number  by  which  to  divide. 

100.  The  Quotient  is  the  result  of  the  division. 

101.  The  Sign  of  Division  is  a  short  horizontal  line,  with 
a  dot  above  and  another  below  it,  -^-,  or  a  short  upright 
curved  line,  ),  and  is  read  divided  by. 

Thus,  40  -*-  8,  or  8)40,  is  read,  forty  divided  by  eight. 

WRITTEN    EXERCISES. 

102.  Copy  and  divide- 

r* •  •         oio   n-  -j     j  Solution.— 2  is  contained  in  8, 

Divisor,  2)8  Dividend.  __r  .  ,       ,      ,.  .' 

9      —  4  times.    Write  4  under  the  divi- 

4  Quotient.       dend  for  the  quotient. 
(2.)  (3.)  (4.)  (5.)  (6.) 

3)18         5)15         4)12  6)30  8)J8 

q)zq  Solution. — 8  is  contained  in  50,  6  times, 

and  2  remain.     Write  6  as  the  quotient,  and 

6,   2  Rem.       t^e  2  as  a  remainder. 

(8.)  (9.)  (10.)  (11.)  (12.) 

7)57  6)55         8)75         9)J7  7)58 

Multiply  the  quotient  by  the   divisor,  and  to   this 


DIVISION.  57 

product  add  the  remainder,  in  each  of  the  last  five 
problems.  If  this  result  equals  the  dividend,  the  work 
is  correct. 

MENTAL   EXERCISES. 

103. — 1.  What  do  you  understand  by  one  half  of  a 
number  ? 

One  of  the  two  equal  parts  into  which  the  number  is  divided. 

2.  What  do  you  understand  by  one  third  of  a  num- 
ber ?     By  one  fourth  of  a  number  ? 

3.  What  do  you  understand  by  two  thirds  of  a  num- 
ber ?     By  two  fourths  of  a  number  ? 

4.  What  part  of  5  is  1  ?     Is  2?     Is  3?     Is  4? 

Solution. — One  is  1  fifth  of  5 ;  2  is  2  times  1  fifth  of  5,  or  2 
fifths  of  5 ;  3  is  3  times  1  fifth  of  5,  or  3  fifths  of  5 ;  and  4  is  4 
times  1  fifth  of  5,  or  4  fifths  of  5. 

5.  What  part  of  6  is  1  ?     Is  2?     Is  3?     Is  4?     Is  5? 

6.  Seven  is  how  many  times  3  ? 

Solution. — Seven  is  2  times  3,  and  1  remains,  which  is  1 
third  of  3.    Therefore,  7  is  2  and  1  third  times  3. 

7.  Nine  is  how  many  times  2?     4?     3?     8? 

8.  Eight  is  how  many  times  6  ? 

9.  Nine  is  how  many  times  6  ?     7  ?     8  ? 

10.  At  8  dollars  a  yard,  how  many  yards  of  velvet 
can  be  bought  for  33  dollars  ? 

11.  At  9  cents  a  pound,  how  many  pounds  of  rice 
can  be  bought  for  47  cents  ? 

12.  15  is  how  many  times  4? 

Solution. — Fifteen  is  3  times  4,  with  a  remainder  3 ;  or  3  and 
3  fourths  times  4. 


58  DIVISION. 

13.  How  many  yards  of  braid,  at  7  cents  a  yard,  can 
be  bought  for  52  cents  ? 

14.  Thirty-eight  is  how  many  times  7  ?     5  ? 

15.  Fifty-nine  is  how  many  times  5  ?     8  ? 

16.  If  9  pieces  of  cloth  cost  95  dollars,  what  is  the 
cost  of  each  piece  ? 

17.  Eighty-three  is  how  many  times  9  ?     7  ? 

18.  What  is  one  fifth  of  52  ?     Of  63  ?     Of  49  ? 

19.  What  is  one  seventh  of  31  ?     Of  57  ?     Of  68  ? 

20.  What  is  one  eighth  of  46  ?     Of  39?     Of  73  ? 

21.  If  75  persons  are  to  cross  a  stream  in  a  boat 
which  will  carry  only  8  persons  at  a  time,  how  many 
persons  will  remain  after  as  many  full  boat-loads  as  pos- 
sible have  crossed  ? 

22.  If  10  bushels  will  fill  a  bin,  how  many  bins  can 
be  filled  from  87  bushels  of  wheat,  and  how  many 
bushels  will  remain  ? 

23.  How  many  plows,  worth  11  dollars  each,  can  be 
bought  with  93  dollars,  and  how  many  dollars  will  be 
left? 

24.  In  one  hundred  and  thirty  eggs  are  how  many 
dozen,  and  how  many  remain  ? 

DEFINITIONS. 

104.  A  Remainder  is  a  part  of  the  dividend  which 
may  remain  undivided. 

105.  An  Integer  is  a  number  composed  of  entire  units 
or  ones. 

106.  A  Fraction,  as  one  half,  two  thirds,  etc.,  is  a 
number  which  represents  one  or  more  of  the  equal  parts 
of  a  unit  or  one. 


DIVISION.  59 

One  half  is  written  J,  one  third  is  written  J,  two 
thirds  are  written  f ,  three  fourths  are  written  f,  and  so 
on.  In  such  expressions  the  number  written  below  the 
line  denotes  the  number  of  parts  into  which  the  unit  is 
divided,  and  the  number  written  above  the  line  denotes 
the  number  of  parts  taken. 

Principles  of  Division. 

107. — 1.  Division  is  the  reverse  of  multiplication. 

2.  The  dividend  is  equal  to  the  product  of  the  integer 
of  the  quotient  multiplied  by  the  divisor,  plus  the  re- 
mainder. 

WRITTEN   EXERCISES. 

108. — 1.  Divide  4315  by  4,  or  separate  4315  into  4 
equal  parts ;  and  prove  the  solution. 

Divisor,  4)4315    Dividend.  Solution.— One  of  the  4 

3  t  equal  parts  of  a  number  is 

107 8^  Quotient         one  fourth  of  that  uumber. 

One  fourth  of  4  thousands  is  1  thousand.  Write  1  in  the 
thousands'  order  in  the  quotient. 

One  fourth  of  3  hundreds  is  0  number  of  hundreds.  Write 
0  in  the  hundreds'  order  in  the  quotient,  and  unite  the  3  hun- 
dreds with  the  1  ten,  making  31  tens. 

One  fourth  of  31  tens  is  7  tens,  with  3  tens  remaining.  Write 
7  in  the  tens'  order  in  the  quotient,  and  unite  the  3  tens  with  the 
5  ones,  making  35  ones. 

One  fourth  of  35  ones  is  8  ones,  with  3  ones  remaining,, 
Write  the  8  in  the  ones'  order  in  the  quotient. 

Write  the  remainder,  3,  over  the  divisor  as  a  part  of  the 
quotient. 

The  3  written  over  4,  or  f,  may  be  regarded  as  indicating 
the  division  of  3  by  4. 

The  result  is  1078f ,  which  is  the  quotient  required. 


60 


DIVISION. 


1078 

J^  Proof. — To  prove  the  work,  multiply  the  integer 

T0T9  of  the  quotient  by  the  divisor,  and  add  to  the  product 

„  the  remainder.    If  the  work  is  correct,  this  result  will 

equal  the  dividend. 

4815 

When  only  the   divisor,  dividend  and  quotient  are 
written,  the  process  is  called  Short  Division. 

Divide,  by  short  division,  explain  and  prove — 


(2.) 

4)3101 

~T75l 

(6.) 
7)300 


(3.) 
5)5163 

103 2 j 

(7.) 
3)975 


(4.) 
7)814 


(8.) 
6)1992 


(5.) 
8)1137 

(9.) 
9)2889 


10.  How  many  is  one  fourth  of  731  apples? 

11.  What  is  the  quotient  of  1363  days  divided  by  2? 

12.  What  is  the  quotient  of  1563  divided  by  10  ? 


10)1563 

156.3 

Or,  156f0 


Solution. — To  divide  by  10,  simply  move  the 
decimal  point  in  the  dividend  one  order  to  the 
left,  because  this  changes  each  figure  to  an  order 
next  lower,  and  makes  the  value  expressed  only 
one  tenth  as  large  as  before. 
The  remainder  is  3,  and  written  at  the  right  of  the  point,  ex- 
presses 3  tenths,  or  written  over  the  divisor,  T\. 
The  result  is  156TV,  which  is  the  quotient  required. 

13.  What  is  the  quotient  of  715  divided  by  100? 

14.  What  is  the  quotient  of  1634  divided  by  10? 

15.  What  is  the  quotient  of  1783  divided  by  1000? 


DIVISION.  61 


MENTAL  EXERCISES. 

109. — 1.  How  many  pencils,  at  7  cents  each,  can  be 
bought  for  56  cents  ? 

Solution. — Since  one  pencil  can  be  bought  for  7  cents,  as 
many  pencils  can  be  bought  for  5G  cents  as  there  are  times  7 
in  56,  which  are  8  times. 

Therefore,  8  pencils  at  7  cents  each  can  be  bought  for  56  cents. 

2.  How  many  barrels  of  flour,  at  8  dollars  each,  can 
be  bought  for  64  dollars  ? 

3.  How  many  pounds  of  sugar,  at  9  cents  a  pound, 
can  be  bought  for  72  cents  ? 

4.  If  you  have  60  dollars,  and  spend  it  at  the  rate 
of  10  dollars  a  week,  how  many  weeks  will  your  money 
last  you  ? 

5.  If  twelve  eggs  are  1  dozen,  how  many  dozen  are 
72  eggs? 

6.  When  pine-apples  are  8  cents  each,  how  many  can 
be  bought  for  64  cents  ? 

7.  When  96  dollars  are  paid  for  12  barrels  of  flour, 
how  much  is  paid  for  each  barrel  ? 

8.  How  many  melons  can  be  exchanged  for  110 
peaches,  at  the  rate  of  11  peaches  for  1  melon? 

9.  How  many  pounds  of  beef,  at  11  cents  a  pound,  can 
be  bought  for  7  9  "cents,  and  how  many  cents  will  remain  ? 

10.  At  12  cents  a  pound,  how  many  pounds  of  sugar, 
and  what  part  of  .a  pound,  can  be  bought  for  109  dollars  ? 

11.  How  many  are  12  times  9,  plus  1  ?  9  times  12, 
plus  1  ? 

12  John  gained  24  dollars  by  purchasing  wood  at  5 
dollars  a  cord  and  selling  it  at  8  dollars  a  cord.  How 
many  cords  did  he  buy  ? 

6 


62  DIVISION. 

WRITTEN  EXERCISES. 

110. — 1.  Divide  672  by  4,  and  prove  the  solution. 

Solution. — 4  is  contained  in  6  hundred^ 

iY18/T)^oV  iaq       *  nun^re<^  times-     Write  1  in  the  order  of 

4yOi^\  loo       hundreds  in  the  quotient,  which  for  con- 

4  venience  is  placed  on  the  right  of  the  divi- 

27 .  dend. 

2 J.  Multiply  4,  the  divisor,  by  the  1  hundred, 

making  4  hundreds,  which  write  under  the 

0/0  6  hundreds,  and  subtract  from  it,  leaving  2 

32  hundreds.     Unite  the  2  hundreds  with  the 

7  tens,  making  27  tens. 
4  is  contained  in  27  tens,  6  tens  times.     Write  the  6  in  the 
order  of  tens  in  the  quotient. 

Multiply  the  divisor  by  the  6  tens,  making  24  tens,  which 
write  under  the  27  tens,  and  subtract,  leaving  3  tens.  Unite 
the  3  tens  with  the  2  ones,  making  32  ones. 

4  is  contained  in  32  ones,  8  times.  Write  the  8  in  the  order 
of  ones  in  the  quotient.  The  result  is  168,  which  is  the  quotient 
required. 

-LUO  Proof. — To  prove  the  work,  multiply  the  quotient 

-4        by  the  divisor.     If  this  product  equals  the  dividend, 

672        tne  work  is  correct. 

When  each  step  of  the  solution  is  written,  the  process 
is  called  Long  Division. 

Divide  and  prove  in  like  manner — 


2.  540  by  5. 
a  896  by  2. 
4.  822  hy  6. 


5.  423  by  3. 

6.  57^by^. 

7.  655  by  5. 

111.  Rule  for  Division.—  Write  the  divisor  at  the  left 
of  the  dividend. 
Divide  the  least  number  of  the  left-hand  orders 


8.  936  by  8. 

9.  796  by  2. 
10.  504  by  7. 


DIVISION.  63 

of  the  dividend  thai  will  contain  the  divisor,  and 
place  the  quotient  at  the  right  of  the  dividend,  in 
long  division,  and  beneath  the  dividend  in  short 
division. 

Multiply  the  divisor  by  this  quotient;  subtract 
the  result  from  that  part  of  the  dividend  which 
was  used;  to  the  remainder  annex  the  next  order 
of  the  dividend,  and  divide  the  number  thus 
formed. 

Proceed  in  the  same  manner  until  all  the  orders 
of  the  dividend  have  been  used. 

Should  there  be  at  last  a  remainder,  write  it,  with 
the  divisor  under  it,  as  a  part  of  the  quotient. 

When  the  divisor  is  1,  with  one  or  more  ciphers  on  the  right, 
move  the  decimal  point  in  the  dividend  as  many  orders  to  the 
left  as  there  are  ciphers  on  the  right  of  the  divisor.  The  orders 
on  the  left  of  the  point  will  be  the  integer  of  the  quotient,  and 
the  orders  on  the  right  the  remainder  expressed  as  a  fractional 
part  of  the  quotient. 

Proof. — Multiply  the  integer  of  the  quotient  by  the  divisor, 
and  add  to  the  product  the  remainder,  if  any.  If  the  work  is 
correct,  this  result  will  equal  the  dividend. 

PMOBZEMS. 

112.  Divide,  explain  and  prove — 

(1.)  (2.)  (3.)  (4.) 

2)1363        8)4602        7)703       4)2060 

(5.)         (6.)        (7.)        (8.) 
14)4651         15)3910       21)443       12)6702 


64  DIVISION. 


113.  How  many  is — 

1.  One  fourth  of  731  apples? 

2.  One  third  of  563  cents? 

3.  One  sixth  of  802  feet? 

4.  One  seventh  of  415  rods? 

5.  One  ninth  of  629  dollars  ? 


What  is  the  quotient  of—, 

6.  6363  days  -=-  11  ? 

7.  2741  pounds  --13? 

8.  1790  dollars --7? 

9.  8000  cents  -=-  25  ? 
10.  4350  yards  ~  9  ? 

114.— 1.  What  is  one  tenth  of  3587? 

2.  If  15  acres  of  land  cost  6090  dollars,  how  many 
dollars  will  one  acre  cost  ? 

3.  How  many  hours  will  it  take  a  train  of  cars  to 
move  1225  miles,  at  the  rate  of  25  miles  per  hour? 

4.  What  number  is  equal  to  1488  --  24  ? 

5.  What  number  is  equal  to  4141  -=-  101  ? 

6.  How  many  times  may  a  31 -gallon  cask  be  filled 
from  a  vat  containing  1929  gallons,  and  how  many 
gallons  will  remain? 

7.  Into  how  many  lots  of  10  acres  each  can  365  acres 
of  land  be  divided,  and  how  many  acres  will  remain  un- 
divided ? 

8.  If  100  feet  of  boards  will  make  a  box,  how  many 
similar  boxes  can  be  made  from  5767  feet  of  boards, 
and  how  many  feet  will  remain  ? 


115.  Test  Questions.— 1.  What  is  division  ?     What  is  the 
dividend  ?    The  divisor  ?    The  quotient  ? 

2.  What  is  the  sign  of  division?    Show  how  it  is  used? 
What  does  it  mean  ? 

3.  What  is  a  remainder?     An  integer?    A  fraction? 

4.  What  is  short  division?     Long  division ?    What  principles 
of  division  are  given  ? 


ME  VIEW.  65 

SECTION  VIII. 

REVIEW. 

116. — 1.  By  what  process  do  you  find  the  product  of 
two  or  more  numbers  ? 

2.  By  what  process  do  you  find  the  quotient  of  one 
number  divided  by  another  ? 

3.  How  do  multiplication  and  division  differ  ? 

4.  If  the  multiplicand  is  dollars,  what  will  be  the  de- 
nomination of  the  product? 

5.  If  the  multiplicand  is  6  and  the  multiplier  4,  how 
many  times  the  multiplicand  is  the  product  ? 

6.  What  does  the  product  express  ? 

7.  If  the  divisor  is  4  dollars  and  the  dividend  24 
dollars,  what  does  the  quotient  express  ? 

8.  When  the  divisor  is  4  and  the  dividend  24  dollars, 
what  part  of  24  dollars  is  the  quotient  ? 

9.  To  what  term  in  multiplication  does  the  dividend 
in  division  correspond  ? 

10.  To  what  terms  in  division  do  the  factors  in  multi- 
plication correspond  ? 

11.  How  many  are  7  times  12,  plus  8  ? 

12.  How  many  are  9  times  11,  minus  7  ? 

13.  When  5  tons  of  coal  cost  35  dollars,  what  will 
7  tons  cost  ? 

Solution. — If  5  tons  cost  35  dollars,  1  ton  will  cost  one-fifth 
of  35  dollars,  or  7  dollars,  and  7  tons  will  cost  7  times  7  dollars, 
which  are  49  dollars. 

14.  When  8  barrels  of  flour  cost  88  dollars,  what  will 
5  barrels  cost  ? 


06  REVIEW. 

15.  James  bought  12  bags  of  meal  for  24  dollars, 
and  Smith  bought  11  bags  at  the  same  rate.  How  many 
dollars  did  Smith  pay  for  his  meal  ? 

16.  If  you  can  earn  56  dollars  in  8  weeks,  how  much 
can  you  earn  in  3  weeks  ? 

17.  If  a  horse  can  trot  at  the  rate  of  24  miles  in  3 
hours,  how  far  can  he  trot  in  2  hours  ? 

18.  If  5  men  can  do  a  piece  of  work  in  6  days,  in 
how  many  days  can  3  men  do  it  ? 

Solution. — If  5  men  Can  do  a  piece  of  work  in  6  days,  1 
man  will  require  5  times  6  days,  or  30  days,  to  do  it,  and  3  men 
will  require  one  third  of  30  days,  or  10  days. 

19.  If  6  men  can  mow  a  field  in  8  days,  in  how  many 
days  can  12  men  mow  it? 

20.  When  8  bushels  of  wheat  will  pay  for  4  yards 
of  cloth  at  4  dollars  a  yard,  what  is  the  value  of  a 
bushel  of  wheat? 

21.  When  4  yards  of  cloth  can  be  bought  for  16  dol- 
lars, how  many  yards  of  cloth  can  be  bought  for  40 
dollars  ? 

Solution.— If  4  yards  can  be  bought  for  16  dollars,  1  yard 
can  be  bought  for  one  fourth  of  16  dollars,  or  4  dollars.  Since 
1  yard  can  be  bought  for  4  dollars,  as  many  yards  can  be  bought 
for  40  dollars  as  there  are  times  4  in  40,  or  10. 

22.  If  5  horses  will  consume  60  bushels  of  oats  in  a 
certain  time,  how  many  horses  will  consume  72  bushels 
in  the  same  time  ? 

23.  When  99  dollars  will  pay  for  11  barrels  of  flour, 
how  many  barrels  can  be  bought  for  63  dollars  ? 

24.  If  you  can  earn  108  dollars  in  12  months,  how 
many  dollars  can  you  earn  in  8  months  ? 


MB  VIEW.  67 

WRITTEN  EXERCISES. 

117. — 1.  A  merchant  bought  12  hogsheads  of  molasses 
it  65  dollars  a  hogshead,  and  sold  them  for  805  dollars. 
How  much  did  he  gain  by  the  transaction  ? 

2.  From  6304  X  15,  subtract  372  --  3. 

3.  If  you  sell  506  hats,  which  cost  you  4  dollars  each, 
for  6  dollars  each,  how  much  will  you  gain  ? 

4.  From  1035  X  6,  take  1060  X  5. 

5.  I  bought  8  horses  at  225  dollars  each,  and  17  yoke 
of  oxen  at  150  dollars  a  yoke.  What  is  the  cost  of  the 
whole?    ■ 

6.  If  a  man's  salary  is  5000  dollars  a  year,  and  his 
expenses  are  225  dollars  a  month,  how  much  can  he  save 
each  month  ? 

7.  A  farmer  has  131  bushels  of  corn,  twice  as  much 
rye,  and  three  times  as  much  wheat.  What  quantity  of 
rye  and  of  wheat  has  he  ? 

8.  When  19  acres  of  land  cost  760  dollars,  how  many 
acres  can  be  bought  for  2180  dollars? 

9.  If  a  quantity  of  provisions  will  last  355  men  25 
days,  how  long  will  it  last  5  men  ? 

10.  When  755  dollars  are  paid  for  5  horses,  how 
much  must  be  paid  for  31  horses  ? 

11.  I  bought  a  farm  containing  120  acres  for  7200 
dollars,  and  sold  it  for  75  dollars  an  acre.  How  much 
did  I  gain  by  the  transaction  ? 

12.  How  much  is  255  X  12  divided  by  51  X  8  ? 

13.  How  many  casks,  holding  31  gallons  each,  can  be 
filled  from  17  hogsheads  of  molasses,  containing  each  93 
gallons  ? 


68  FACTORING. 

SECTION    IX. 

FACTORS. 

118,-1.  How  many  ones  in  2  ?     In  3  ?     In  10  ? 

2.  What  two  integers,  then,  multiplied  together,  will 
produce  2  ?     3  ?     10  ?     What  15?     21  ? 

3.  What  integers,  other  than  itself  and  1,  multiplied 
together,  will  produce  14  ?     22  ?     39  ? 

4.  What  integers,  other  than  itself  and  1,  will  divide 
14  without  a  remainder  ?     22  ?     39  ? 

5.  What  integers,  multiplied  together,  will  produce  9  ? 
10?     16?     18? 

6.  WThat  are  all  the  integers  which  will   divide  9 
without  a  remainder  ?     10?     16?     18? 

7.  Name  some  number  which  is  not  the  product  of 
any  integers  except  itself  and  1. 

8.  Name  some  number  which  is  the  product  of  other 
integers  than  itself  and  1. 

DEFINITIONS. 

119.  The  Factors  of  a  number  are  the  integers  which, 
being  multiplied  together,  will  produce  that  number. 

120.  A  Prime  Number  is  an  integer  that  has  no  factor 
except  itself  and  1. 

121.  A  Composite  Number  is  an  integer  that  has  other 
factors  besides  itself  and  1. 

122.  The  Prime  Factors  of  a  number  are  the   prime 
numbers  that  are  factors  of  that  number. 

Since  1  is  a  factor  of  all  numbers,  in  naming  the  prime  fac- 
tors of  numbers  it  need  not  be  given. 


FACTORING.  69 

123,  Factoring  is  the  process  of  finding  the  factors  of 
composite  numbers. 

124—1.  What  are  the  prime  factors  of  12?    14?    16? 

2.  What  are  the  prime  factors  of  15  ?     18  ?     20  ? 

3.  What  are  the  prime  factors  of  21  ?     28  ?     30  ? 

4.  What  are  the  prime  factors  of  33  ?     45  ?     49  ? 

5.  What  are  the  prime  factors  of  27  ?     42  ?     44  ? 

6.  What  are  the  prime  factors  of  36  ?     40  ?     54  ? 

7.  Name  the  prime  numbers  from  1  to  19.     From  19 
to  29. 

8.  Name  the  prime  numbers  from  31  to  41.     From 
41  to  83. 

9.  Name  the  composite  numbers  from  4  to  20.     From 
24  to  36.     From  36  to  48.     From  48  to  60. 

125.  Principle. — Every  composite  number  is  equal  to 
the  product  of  all  its  prime  factors. 

WRITTEN  EXERCISES. 

126. — 1.  What  are  the  prime  factors  of  90? 

2)90  Solution.— Dividing  90  by  the 

n  jyz  prime  number  2,  we  have  as  factors 

^  2  and  the  composite  number  45. 

3)15  Dividing  45  by  the  prime  num- 

5  bar  3,  we  have  as  factors  3  and  15. 

on— 4>V*V*V*        dividing  again  by  3,  we  have  as 
VU -    ^AJAJAO         factors  3  and  5    which  are  prime 

numbers.      Hence,  90  =  2X3X3X5,  and  the  prime  factors 
of  90  are  2,  3,  3  and  5. 

2.  What  are  the  prime  factors  of  84  ? 

3.  What  are  the  prime  factors  of  75  ? 

4.  What  are  the  prime  factors  of  96  ? 


70 


FACTORING. 


127,  Rule  for  finding  the  Prime  Factors  of  a  Number.— Divide 
the  given  number  by  any  prime  number  greater 
than  1  that  will  divide  it  without  a  remainder 
and  the  quotient,  if  composite,  in  the  same  man- 
ner; and  so  proceed  until  a  quotient  is  obtained 
which  is  a  prime  number. 

The  last  quotient  and  the  several  divisors  are  the 
prime  factors. 

PROBLEMS. 

128.  "What  are  the  prime  factors  of— 


1.  95? 

4.  122? 

7.  108? 

10.  148? 

2.  63? 

5.  116? 

8.  200? 

11.  210? 

3.  92? 

6.  184? 

9.  728? 

12.  735? 

SECTION   X. 

DIVISORS. 

129. — 1.  What  integers  will  divide  15  without  a  re- 
mainder?    21?     35? 

2.  What  integers  will  divide  16  without  a  remainder? 
27  ?     42  ? 

3.  What  integers  will  divide  44  without  a  remainder  ? 
77?     81? 

4.  What  factors  have  6  and  24  in  common?  15  and  20  \ 

5.  What  factors  have  22  and  25  in  common  ? 

6.  What  prime  factors  have  12  and  18  in  common? 

7.  Wrhat  is  the  greatest  factor  common  to  12  and  18  ? 

8.  What  is  the  product  of  the  prime  factors  common 
to  12  and  18? 


FACTORING.  71 

9.  What  is  the  product  of  the  prime  factors  common 
to  8  and  24  ? 

10.  What  is  the  greatest  factor  common  to  3  and  24  ? 

DEFINITIONS. 

130.  A  Divisor,  or  Measure,  of  a  number  is  any  factor 
of  that  number. 

131.  A  Common  Divisor  of  two  or  more  numbers  is 
any  factor  common  to  those  numbers. 

132.  The  Greatest  Common  Divisor  of  two  or  more  num- 
bers is  the  greatest  factor  common  to  those  numbers. 

133.  Principle. — The  greatest  common  factor,  or  divi- 
sor, of  two  or  more  numbers  is  equal  to  the  product  of 
all  the  common  prime  factors  of  those  numbers. 

WRITTEN  EXERCISES. 

134. — 1.  What  is  the  greatest  common  factor  or  di- 
visor of  330  and  550  ? 

330  =  2  X  3  X  5  X  11  Solution.  — Find   the  prime 

QfiO  =  2  X  5  X  5  X  11       Actors  of  the  numbers. 

^y  ry  1 1  =  1  in  ^ne  Prime  factors  common  to 

the  numbers  are  2,  5  and  11,  and 
their  product  is  110.  Hence,  the  greatest  common  factor  or 
divisor  is  110. 

2.  Find  the  greatest  common  factor  of  27  and  36. 

3.  Find  the  greatest  common  factor  of  42  and  35. 

4.  Find  the  greatest  common  divisor  of  14, 35  and  63. 

135.  Rule  for  finding  the  Greatest  Common  Divisor.— Find 
the  prime  factors  common  to  the  given  numbers, 
anal  the  product  of  those  factors  will  be  the  greatest 
co /union  divisor  of  the  numbers. 


72  FACT0R1XG. 

PROBLEMS. 

136.  What  is  the  greatest  common  factor  or  divisor  of—* 
1.  26  and  39?    14.  18  and  96?     !  7.  4,  6  and  18? 


2.  17  and  51?    \  5.  45  and  300? 

3.  27  and  63?       6.  21  and  105? 


8.  12,  30  and  84? 

9.  18,  54  and  72? 


10.  What  is  the  length  of  the  longest  pole  which  will 
exactly  measure  130,  150  or  170  feet? 


SECTION    XI. 
MULTIPLES. 


137. — 1.  Name  the  numbers  from  3  to  15  that  will 
contain  3  without  a  remainder. 

2.  What  are  some  of  the  numbers  that  are  an  exact 
number  of  times  3  ? 

3.  What  are  some  of  the  numbers  that  are  an  exact 
number  of  times  5  ?     Of  times  8  ?     Of  times  9  ? 

4.  Of  what  integers  is  12  an  exact  number  of  times? 

5.  Name  some  number  that  will  contain  either  3  or  4 
an  exact  number  of  times. 

6.  Name  some  dividend  that  will  exactly  contain  2 
and  6.     5  and  7.     6  and  9. 

7.  What  is  the  least  dividend  that  will  contain  both 
10  and  15  an  exact  number  of  times? 

8.  What  are  all  the  prime  factors  of  10  and  15  ? 

9.  What  are  prime  factors  of  the  least  number  that 
will  exactly  contain  both  10  and  15? 


FACTORING.  73 

DEFINITIONS. 

138.  A  Multiple  of  a  number  is  any  number  which  it 
will  divide  without  a  remainder. 

139.  A  Common  Multiple  of  two  or  more  numbers  is 
any  number  which  each  of  those  numbers  will  divide 
without  a  remainder. 

140.  The  Least  Common  Multiple  of  two  or  more  num- 
bers is  the  least  number  that  each  of  those  numbers  will 
divide  without  a  remainder. 

141.  Principle. — The  least  common  multiple  of  two 
or  more  numbers  is  the  least  number  that  contains  all  the 
prime  factors  of  those  numbers. 

WRITTEN  EXERCISES. 

142. — 1.  What  is  the  least  common  multiple  of  4,  12 
and  30? 

/  —  2  X  2.  Solution. — Find  the  prime  fac- 

1Q  =  Q  V  I?  X  ?  *ors  °^  ^ne  given  numbers. 

to* ^y  ^y   -  The  multiple  of  4  must  contain 

"\  '  the  prime   factors   2   and    2 ;    the 

%  X  2  X  3  X  o  =  60.  muitiple  of  12,  the  additional  prime 
factor  3 ;  and  the  multiple  of  30,  the  additional  prime  factor  5. 
These  factors  are  2,  2,  3  and  5,  and  their  product  is  60.  There- 
fore, the  least  common  multiple  of  4,  12  and  30  is  60. 

2.  Find  the  least  common  multiple  of  8  and  12. 

3.  Find  the  least  common  multiple  of  4,  6  and  20. 

143.  Rule  for  finding  the  Least  Common  Multiple.— First  find 
the  prime  factors  of  the  given  numbers.  The  prod- 
uct of  these  different  prime  factors,  each  factor 
being  taken  the  greatest  number  of  times  it  occurs 
in  any  of  the  numbers,  will  be  their  least  common 
}  multiple. 

7 


FACTORING. 
PROBLEMS. 

144.  What  is  the  least  common  multiple  of — 


7.  6,  8  and  10? 

8.  12,  15  and  16  \ 

9.  16,  35  and  70? 


1.  16  and  18?  4.  14  and  21  ? 

2.  36  and  108  ?       5.  42  and  63  ? 

3.  24  and  84?  6.  19  and  57? 
10.  What  is  the  smallest  sum  of  money  for  which 

you  can  purchase  a  number  of  pears  at  either  5  cents, 
10  cents  or  12  cents  each? 


SECTION    XII. 
CANCELLATION. 


145. — 1.  What  is  the  quotient  of  60  divided  by  15  ? 

2.  What  is  the  quotient  of  one-third  of  60  divided 
by  one-third  of  15? 

3.  What  is  the  quotient  of  one-fifth  of  60  divided  by 
one-fifth  of  15? 

4.  What  common  factors  have  60  and  15  ? 

5.  If  you  take  out  from  60  the  factors  common  to  15, 
what  factor  will  remain  ? 

6.  What  is  the  quotient  of  2  X  7  X  3  divided  by 
2X7? 

7.  If  you  strike  out  from  42  and  14  all  common 
factors,  and  divide  the  remaining  factors,  what  is  the 
quotient  ? 

DEFINITION. 

146.  Cancellation  is  the  process  of  shortening  compu- 
tations by  striking  out  equal  factors  from  the  dividend 
•and  divisor,  and  using  only  the  remaining  factors. 


FACTORING.  75 

147.  Principle. — Striking  out  equal  factors  from  both 
dividend  and  divisor,  or  dividing  them  by  the  same  num- 
ber j  does  not  change  their  quotient 

WRITTEN  EXERCISES. 

148—1.  Divide  11  X  3  X  2  by  11  X  2. 

1  1  Solution.— Write  the 

iHr  X  ^  X  i1     1X3X1  dividend  over  the  divisor. 

~r/T~77  ,         v —  3  Divide  both  dividend 

11  1  X  7 

and  divisor  by  11  and  2, 

^  by  canceling  or  striking 

out  those  common  factors  in  both,  leaving  1X3X1-5-1X1  =  3. 

When  a  factor  is  canceled,   1  remains,  and  if  not  written  is 

understood. 

2.  Divide  150  by  30. 

Solution. — Since    the    dividend  and  divisor 
8 j  0)15\  0     have  10  as  a  common  factor,  cancel  it  by  striking 
g  off  the  cipher  from  the  right  of  both,  leaving 

3)15,  which  equals  5. 

3.  Divide  13  X  11  X  7  by  13  X  3.     1001  by  13  X  3. 

4.  Divide  2500  by  500.     360  by  10  X  9. 

149.  Rule  for  Cancellation.— Cancel  in  the  dividend 
and  divisor  factors  common,  to  both,  and  then 
divide  the  product  of  the  remaining  factors  of  the 
dividend  by  the  product  of  the  remaining  factors 
of  the  divisor. 

PROBLEMS. 

150.  Divide — 


1.  48X3x5by8X7X5. 

2.  50X5X3  by  15  X  10. 

3.  75XllX2byllX5X2. 


4.  31  X  25  by  7  X  5  X  5. 

5.  4X13X3  by  39X2. 

6.  81X8X7  by  27  X  28. 


7.  How  many  tons  of  coal  at  9  dc  liars  a  ton  can  be 
exchanged  for  81  barrels  of  apples  at  3  dollars  a  barrel? 


76  FACTORING. 

8.  How  many  tons  of  hay  at  24  dollars  a  ton  can  lx 
exchanged  for  8  thousand  feet  of  boards  at  15  dollars 
a  thousand  ? 

9.  Divide  11461  by  1400,  using  the  factors  100  and  14. 

141  00)114)  61(8^       Solution.— Strike  off  two  orders  of 

j  jp  figures  from  the  right  of  the  dividend 

and  divisor,  which  divides  each  by 

261  100,  and  gives  for  a  new  dividend  114 

hundreds,  and  61  remaining,  and  for  a  new  divisor,  14  hundreds. 

The  new  dividend  114,  divided  by  the  new  divisor  14,  gives 

a  quotient  8  and  a  remainder  2,  which  is  2  hundreds.     Annexing 

to  this  remainder  the  first  remainder,  61,  gives  261,  the  entire 

remainder,  and  the  entire  quotient  is  8T2?Vo- 

10.  Divide  450  by  70,  using  the  factors  10  and  7. 

11.  Divide  11400  by  600,  using  the  factors  100  and  6. 

12.  If  you  should  earn  1350  dollars  in  300  days, 
how  much  would  you  earn  each  day  ? 


151.  Test  Questions. — 1.  What  are  the  factors  of  a  num- 
ber? What  is  a  prime  number?  A  composite  number?  What 
are  the  prime  factors  of  a  number?  What  is  factoring?  A 
principle  of  factoring? 

2.  What  is  a  divisor,  or  measure,  of  a  number?  A  common 
divisor  of  two  or  more  numbers?  The  greatest  common  divisor 
of  two  or  more  numbers?  A  principle  of  the  greatest  common 
factor  or  divisor? 

3.  When  is  a  number  a  multiple  of  another?  A  common 
multiple  of  two  or  more  numbers?  The  least  common  multiple 
of  two  or  more  numbers?  What  is  a  principle  of  the  least 
common  multiple  of  two  or  more  numbers? 

4.  What  is  cancellation  ?    What  is  a  principle  in  cancellation  ? 


FRACTIONS. 


77 


SECTION    XIII. 


NOTATION  OF  FRACTIONS. 

152. — 1.  When  a  cake  is  divided  into  two  equal  parts, 
what  is  each  of  the  parts  called  ? 

2.  Into  how  many  halves  can  a  cake  be  cut  ? 

3.  When  an  apple  is  divided  into  three  equal  parts, 
what  is  each  of  the  parts  ?  What  are  two  of  the  parts  ? 
How  many  thirds  in  an  apple  ? 

4.  When  an  orange  is  cut  into  four  equal  parts, 
what  is  each  of  the  parts?  What  are  two  of  the 
parts  ?  Three  of  the  parts  ?  How  many  fourths  in  an 
orange  ? 

5.  If  a  cake  be  divided  into  five  equal  parts,  what 
is  each  of  the  parts?  What  are  two  of  the  parts? 
Three  of  the  parts  ?     Four  of  the  parts  ? 

7* 


One  half  is  written  J. 

One  third  "  J. 

One  fourth        "  \. 
One  fifth           " 


78  FRA  CTIONS. 

6.  Into  how  many  halves  can  any  thing  be  divided  ? 
Into  how  many  thirds  ?     Fourths  ?     Fifths  ?     Sixths  ? 

7.  What  is  meant  by  one  half  of  any  thing  ?  By  one 
third  of  any  thing  ?  By  two  thirds  ?  By  one  fourth  ? 
By  two  fourths  ?     By  three  fourths  ? 

8.  Which  are  the  smaller  parts  of  any  thing — halves 
or  thirds  ?  Halves  or  fourths  ?  Thirds  or  fourths  ? 
Halves  or  sixths  ?    Thirds  or  sixths  ?    Fourths  or  sixths  ? 

9.  Halves,  thirds,  fourths,  etc.,  are  expressed  by 
figures,  as  follows — 

Two  thirds  are  written  f . 

Three  fourths  u     f . 

Four  fifths  "     4 . 

5.      Six  sevenths  "     f . 

One  twelfth      "  TV.  j  Nine  eighths  "     f . 

One  fifteenth     "  TV  I   Eleven  sixteenths      "    yg-. 

One  twentieth  "  -^o  •  |   Three  twenty-fourths"    ^V 

10.  How  many  halves  of  1  does  \  express?  How 
many  thirds  of  1  does  f  express  ?  How  many  fourths 
of  1  does  f  express  ? 

1 1 .  What  does  the  figure  4  under  the  dividing  line  in 
the  expression  f  denote  ? 

The  number  of  equal  parts  into  which  the  unit  1  is  divided. 

12.  What  does  the  figure  3  above  the  dividing  line  in 
the  expression  f  denote  ? 

The  number  of  equal  parts  of  the  unit  1  which  are  taken. 

13.  How  is  the  expression  3^  read? 

14.  In  the  expression  Z\y  what  expresses  the  integer  ? 
What  the  fraction  ? 

15.  In  the  expression  7f,  what  expresses  the  integer? 
What  the  fraction  ? 


FRACTIONS. 


79 


DEFINITIONS. 

153.  A  Fraction  is  a  number  which  represents  one  or 
more  of  the  equal  parts  into  which  a  unit  is  divided. 

154.  The  Denominator  of  a  fraction  is  the  number 
which  shows  into  how  many  equal  parts  the  unit  is 
divided. 

155.  The  Numerator  of  a  fraction  is  the  number  which 
shows  how  many  equal  parts  are  taken. 

156.  The  Terms  of  a  fraction  are  its  numerator  and 
denominator. 

157.  A  Mixed  Number  is  a  number  expressed  by  an 
integer  and  a  fraction. 


WRITTEN  EXERCISES. 

158.  Write  and  read — 


1          6 
I.      y. 

7     10 

'  •     19' 

13.  V. 

19.  4|. 

9    -8- 
4.    ii- 

8.  ti- 

14.  V- 

20.  7f 

o      l  8 
»•    ft' 

Q     1A 
V.    i6. 

15.  V4- 

21.  21*. 

A       2 
4-    TV- 

10.  Tv 

16.   W- 

22.  7Ji. 

r     1 6 
0.     2T- 

11.  If. 

1 '  •    i i- 

.Zo.     t/t/y"Q". 

a    1 2 

0.    yy. 

12.  TV 

18.  if* 

24.  106T87. 

159.  Write  in  figures — 

1.  Three  fifths. 

9. 

Two  twenty-firsts. 

2,  Four  sevenths. 

10. 

Five  thirty-seconds. 

3.  Five  eighths. 

11. 

Nineteen  eighths. 

4.  Six  tenths. 

12. 

Thirty-one  ninths. 

5.  Nine  fourths. 

13. 

Six  and  six  elevenths. 

6.  Five  sixths. 

14. 

Twelve  and  four  fifths. 

7.  Four  ninths. 

15. 

Twenty-one  and  one  fourth. 

8.  Two  elev 

enths. 

16. 

Nineteen  anc 

1  one  thirteenth 

80 


FRACTIONS. 


VALUE  OF   A   FRACTION. 


160. — 1.  If  you  should  cut  1  apple  into  halves,  how 
many  halves  would  there  be  ? 

2.  If  you  should  cut  5  apples  into  halves,  how  many 
halves  would  there  be  ?  One  half  of  5  apples  is  how 
many  halves  of  1  apple  ? 

.3.  How  many  apples  are  one  half  of  5  apples?  5 
halves  are  what  part  of  5  apples  ? 

4.  Is  |  greater  or  less  than  1  ?  Is  f  greater  or  less 
than  1  ?     What  is  the  value  of  f  in  ones  ?     Of  \°  ? 

5.  Is  f  greater  or  less  than  1  ?  What  is  the  value  of 
|  in  ones  ?     Of  f  ? 

6.  Considered  as  an  expression  of  division,  what  is 
the  value  of  f?     Of  V? 

7.  Considered  as  an  expression  of  division,  what  is 
the  value  of  V  ?     Of  V  ? 


FRACTIONS. 


81 


DEFINITIONS. 

161.  A  Proper  Fraction  is  a  fraction  whose  numerator 
is  less  than  its  denominator. 

162.  An  Improper  Fraction  is  a  fraction  whose  nume- 
rator is  not  less  than  its  denominator. 

163.  Reduction  of  Fractions  is  the  process  of  changing 
their  form  or  denomination  without  changing  their  value. 

164.  Principle. — The  value  of  a  fraction  is  the  quo- 
tient arising  from  the  division  of  the  numerator  by  the 
denominator. 


WRITTEN  EXERCISES. 

165.  Write  and  name  the  following  fractions — » 


1.  f. 

4.    %. 

7     11 

10.  ,V, 

2.1. 

5.  TV 

8.  Vs- 

11.  ¥■ 

<j.  f. 

6.    *. 

9.  tV 

12.  V8' 

lat  is  the  value  of — 

13.  y? 

16.  V? 

19.  ¥? 

22.  V8? 

14.   V5? 

17.   f? 

20.  V? 

23.  %5? 

15.  ¥? 

18.  y? 

2i.  y»? 

24.  V? 

166.  Test  Questions. — 1.  What  is  a  fraction?  What  is 
the  denominator  of  a  fraction?  The  numerator  of  a  fraction? 
What  are  the  terms  of  a  fraction?  Take  some  fraction  and 
show  what  are  its  terms. 

2.  What  is  a  mixed  number?  Take  some  mixed  number 
and  show  what  expresses  the  integer  and  what  the  fraction. 

3.  What  is  a  proper  fraction?  An  improper  fraction?  What 
is  reduction  of  fractions?     What  is  the  value  of  a  fraction? 


82 


FRACTIONS, 


SECTION  XIV. 
REDUCTION  OF  FRACTIONS. 

CASE    I. 

Integers  or  Mixed  Numbers  Reduced  to  Fractions. 

167—1.  If  1  apple 
be  cut  into  halves,  how 
many  halves  will  there 
be? 

2.  In  2  apples,  how 
many  halves  of  1 
apple? 

Solution. — Since  in  1 
apple  there  are  2  halves, 
in  2  apples  there  must  be 
2  times  2  halves,  which 
are  4  halves. 

3.  How  many  halves  are  there  in  3  ?     In  5  ? 

4.  In  1  orange,  how  many  thirds  of  1  orange  ? 
many  fourths  of  1  orange? 

5.  In  1  dollar,  how  many  fifths  of  1  dollar  ? 
dollars?     In  11  dollars? 

6.  In  3^  apples,  how  many  halves  of  1  apple  ? 

Solution. — Since  in  1  apple  there  are  2  halves,  in  3  apples 
there  must  be  3  times  2  halves,  which  are  6  halves ;  6  halves  and 
1  half  are  7  halves.     Hence,  in  3  J  apples  there  are  J  of  1  apple. 

7.  If  you  had  5^  dollars  to  give  some  poor  boys,  to 
how  many  could  you  give  |  of  1  dollar  each  ? 

8.  In  9^  dollars,  how  many  halves  ?  In  6f ,  how  many 
thirds  ?     In  7f ,  how  many  fourths  ? 


How 


In  6 


FRA  CTIONS. 


83 


9.  In    4j,   how    many    fifths  ?     In     lOf ,  how  many 
eighths  ?     In  3T5T,  how  many  elevenths  ?  . 

10.  How  may  a  number  of  ones  be  changed  to  halves  ? 
To  ninths?     To  twelfths? 


WRITTEN  EXERCISES. 

168. — 1.  Change  9  to  sixths. 

6  sixths  Solution. — Since  in  1  there  are  6  sixths, 

9  in  9  there  must  be  9  times  6  sixths,  which  are 

5j*ixtha=%      54  sixths  =  V- 

6 

2.  Change  19  to  fourths.     25  to  tenths. 

3.  Reduce  65  to  sevenths.     73  to  eighths. 

4.  Change  8|  to  ninths. 

8'U   —8+  g 

g=^2  ninths  Solution. — 8|  is  equivalent  to  8  +  f . 

K  8  is  equal  to  72  ninths,  and  72  ninths 

plus  5  ninths  are  77  ninths  =  y . 


77  ninths  - 


77 
9 


5.  Reduce  13|  to  fifths.     19T8T  to  elevenths. 

6.  Reduce  27-§-  to  thirds.     57T53  to  thirteenths. 

169.  Rule  for  Reduction  of  Integers  or  Mixed  Numbers  to  Im- 
proper Fractions.— Multiply  the  integer  by  the  denomi- 
nator, and,  if  there  be  a  fractional  part,  add  its 
numerator  to  the  product.  This  result,  written  over 
the  denominator,  will  be  the  required  fraction. 


PROBLEMS. 

170.  Reduce  to  improper  fractions — 

4.  14f 

5.  16* 

6.  19* 


1. 

2. 
3. 


4*. 


71- 
11* 


7.  Blf 

8.  20TV 

9.  31* 


10.  61i 

11.  13if. 

12.  16tV 


84 


FRACTIONS. 


CASE    II. 

Fractions  Reduced  to  Integers  or  Mixed  Numbers, 

171.— 1.  In  5  halves 
of  an  apple  are  how 
many  apples  ? 

Solution.  —  Since  2 
halves  of  an  apple  equal 
1  apple,  5  halves  of  an 
apple  must  equal  as  many 
apples  as  2  halves  are  con-  - 
tained  times  in  5  halves, 
which  are  2}  times. 

Hence,  in  5  halves  of  an 
apple  there  are  2J  apples. 

2.  How  many  oranges  will  be  required  to  give  7  boys 
half  an  orange  each  ? 

3.  If  John  can  gather  1  third  of  a  bushel  of  berries 
in  1  day,  how  many  thirds  of  a  bushel  can  he  gather  in 
6  days  ?     How  many  bushels  are  6  thirds  of  a  bushel  ? 

4.  If  a  man  can  spend  1  fourth  of  a  dollar  in  1  day, 
how  many  dollars  can  he  spend  in  8  days  ? 

5.  How  many  ones  in  ^  ?     In  V  ?     In  V5  ?     In  t2  ? 

6.  How  many  ones  in  Y  ?     In  V  ?     In  W     In  V  ? 

WRITTEN   EXERCISES. 

172. — 1.  Change  L^  to  an  equivalent  integer. 

Solution. — Since  4  fourths  equal  one,  112  fourths 
■—  =  28     niust  equal  as  many  ones  as  4  is  contained  times  in 
112,  which  are  28  times.     Hence,  *£*  =  28. 

2.  Change  ^r^  to  an  equivalent  integer. 

3.  Change  -f 4  to  an  equivalent  integer. 


FRACTIONS. 


85 


4.  Change  -^J1  to  an  equivalent  mixed  number. 

Solution. — Since  5  fifths  equal  one,  £73  fifths 

^P  =  11  At     must  equal  as  many  ones  as  5  is  contained  times 

in  573,  which  is  114|  times.     Hence,  *£*  =  114J, 

5.  Change  ^  to  a  mixed  number. 

6.  Change  2T2/  to  a  mixed  number. 

173.  Rule  for  Reduction  of  an  Improper  Fraction  to  an  Integer 
or  Mixed  Number.— Divide  the  numerator  of  the  frac- 
tion by  the  denominator. 

Fit  OB  L  EMS. 

171.  Change  to  integers  or  mixed  numbers — 


1  5  4 
11  • 


4.  HK 

7.  if* 

10. 

1  89 
9 

5.  H1. 

Q       589 
0.      Y9"  . 

11. 

9  13 
~3~ 

6.  HK 

Q      o_l_7 
*•      2  1   • 

12. 

266 
ST 

1. 

2.  H*. 

3.  H*. 


13.  What  is  the  value  of  *$*■  ?     Of  H*  of  a  yard  ? 

C^lSTS   III. 

Fractions  Reduced  to  Higher  or  Lower  Terms. 

175. — 1.  If  an  orange  be  divided  into  two  equal  parts, 
and  each  of  these 
parts  be  divided  into 
two  equal  parts,  what 
part  of  the  orange  is 
one  of  the  pieces  ? 

2.  One  half  is  equal 
to  how  many  fourths  ? 

Solution. — Since  one 
equals  4  fourths,  1  half 
mu3t  equal  one  half  of  4 
fourths,  or  2  fourths. 

a 


86  FRACTIONS. 

3.  If  you  divide  an  orange  into  two  equal  parts,  and 
each  of  these  parts  be  divided  into  three  equal  parts, 
what  part  of  the  orange  will  one  of  the  pieces  be  ? 

4.  One  half  is  equal  to  how  many  sixths  ?     Eighths  ? 

5.  In  what  equivalent  fractions  can  halves  be  ex- 
pressed ? 

In  fourths,  sixths,  eighths,  tenths,  etc. 

6.  The  denominators  of  these  fractions  are  multiples 
of  what  number  ? 

Of  2,  the  denominator  of  \. 

7.  If  an  apple  be  divided  into  three  equal  parts,  and 
each  of  these  parts  into  two  equal  parts,  what  part  of 
the  apple  will  one  of  these  pieces  be  ? 

8.  One  third  is  equal  to  how  many  sixths  ?     Ninths  ? 

9.  In  what  equivalent  fractions  can  thirds  be  ex- 
pressed ? 

10.  The  denominators  of  thirds,  sixths,  twelfths,  etc., 
are  multiples  of  what  number  ? 

11.  The  sixths  of  a  number  are  how  many  times  the 
thirds  ?     The  twelfths  are  how  many  times  the  thirds  ? 

12.  If  you  multiply  both  terms  of  \  by  2,  what  have 
you  ?  If  you  multiply  both  terms  of  -J-  by  4,  what 
have  you  ? 

DEFINITION". 

176.  A  fraction  is  reduced  to  Higher  Terms  when  its 
numerator  and  denominator  are  changed  to  higher 
numbers  without  changing  its  value. 

177.  Principle. — Multiplying  both  terms  of  a  fraction 
by  the  same  number  does  not  change  the  value  of  the 
fraction. 


FRACTIONS.  87 

WRITTEN   EXERCISES. 

178. — 1.  Reduce  f  to  twenty- fourths. 

Solution. — Since    24,    the    required   de- 
5  X  /        £0       nominator,  is  4  times  as  large  as  6,  the  given 

= denominator,  we  multiply  both  terms  of  the 

o  A  4        44       given  fraction  by  4,  which  gives  |f,  the  frac- 
tion required. 

2.  Reduce  \  to  forty-eighths.     t3q-  to  sixty-fourths. 

3.  Reduce  T5T  to  fifty-fifths.     -^  to  seventy-fifths. 

179.  Rule  for  Reduction  of  Fractions  to  Higher  Terms.— 
Multiply  both  terms  of  the  fraction  by  such  a  num- 
ber as  will  give  the  required  denominator, 

PROBLEMS. 

180.  Reduce— 


1.  -J  to  seventy-seconds. 

2.  \  to  sixtieths. 

3.  |  to  eighty-firsts. 


4.  -^  to  sixty-fifths. 

5.  Yw  to  fifty -sevenths. 

6.  -£t  to  one-hundred-eighths. 


case  rv\ 
Fractions  Reduced  to  Lower  Terms. 
181. — 1.  How  many  fourths"  are  equal  to  f  ? 

Solution. — Since  2  eighths  equal  1  fourth,  6  eighths  must 
equal  as  many  fourths  as  2  eighths  are  contained  times  in  6 
eighths,  which  are  3  times.     Hence  }  =  f. 

2.  How  many  fifths  are  equal  to  -^  ?     To^? 

3.  How  many  thirds  are  equal  to  f  ?     To  -jy  ? 

4.  Express  2T  in  parts  two  times  as  great. 

5.  Express  ^  by  a  fraction  having  a  denominator 
one  ninth  as  large. 

6.  Reduce  2T  to  thirds,    ^f-  to  sevenths,     £f  to  eighths. 


SS  FRACTIONS. 

7.  Reduce  ff  to  fourths,     ty  to  ninths.     \%  to  tentha 

8.  yf  is  equal  to  f.  By  what  number  must  you 
divide  both  the  numerator  and  denominator  of  \\  to  ex- 
press the  same  value  by  f  ? 

9.  To  what  fraction  having  smaller  or  lower  terms 
may  f  be  changed  ?     May  \\  ? 

10.  What  common  factors  greater  than  1  have  the 
terms  of  f  ?     Have  the  terms  of  \\  ? 

11.  When  all  the  factors  greater  than  1  common  to 
the  numerator  and  denominator  of  \\  are  cancelled, 
what  will  be  the  fraction  ? 

DEFINITION. 

182.  A  fraction  is  in  its  Lowest  Terms  when  its  nume- 
rator and  denominator  have  no  common  factor  greater 
than  1. 

183.  Principle. — Dividing  both  terms  of  a  fraction  by 
the  same  number  does  not  change  the  value  of  the  fraction. 

WRITTEN   EXERCISES. 

184. — 1.  Reduce  ff  to  its  lowest  terms. 

Solution. — The 
factors  of  the  nu- 
merator 20  are  5, 
2  and  2 ;  the  factors  of  the  denominator  24  are  3,  2,  2  and  2. 
Since  the  fraction  in  its  lowest  terms  can  have  no  factor  greater 
than  1  common  to  both  its  terms,  cancel  the  factors  2  and  2, 
leaving  3^2  =  f,  which  is  the  fraction  in  its  lowest  terms. 

2.  Reduce  ff  to  its  lowest  terms,  ^-f  to  its  lowest 
terms. 

3.  Reduce  ff  to  its  lowest  terms.  ff  to  its  lowest 
terms. 


20 

5X&X-2 

5 

5 

n 

3X-&X3-X  2 

3X2 

6 

FRACTIONS. 


89 


185.  Rule  for  Reduction  of  Fractions  to  their  Lowest  Terms.— 
Cancel  in  both  terms  of  the  given  fraction  all  com- 


PROBLEMS. 

186.  Reduce  to  their  lowest  terms — 

i.  n. 

3.  If. 

5-  fr- 

7    -£- 
f .    54. 

2.  n. 

4.  If. 

a      36 

"•    T0  8~ 

Q         21 

°-    ToT 

CASE    V. 

Fractions  Reduced  to  a  Common  Denominator. 

187. — 1.  William  has  ^  of  a  dollar,  and  John  has  ||. 
How  many  fourths  of  a  dollar  has  each  ? 

Solution. — 1  half  of  a  dollar  is  equal  to  2  fourths  of  a  dollar, 
and  12  sixteenths  of  a  dollar  is  equal  to  3  fourths  of  a  dollar. 
Hence,  William  has  f  of  a  dollar,  and  John  §  of  a  dollar. 

2.  A  father  gave  to  one  of  his  sons  ^  of  a  pine-apple, 
and  to  another  -f .  How  many  fifteenths  of  a  pine-apple 
did  each  receive  ? 

3.  Change  £  and  f  to  eighteenths, 
teenths. 

4.  Reduce 
denominator. 

5.  Reduce 
denominator. 

6.  Reduce  f  and  £  to  fractions  having  a  common 
denominator.  To  fractions  having  the  least  common 
denominator. 

7.  What   is   a  common    multiple  of   4    and    6,    the 


and 


and 


f  and  \  to  four- 
to  fractions  having  the  same 
to  fractions   having   the   same 


denominators  of  £  and  £  ? 


What  is  their  least  common 


multiple  ? 


90  'FRACTIONS, 

8.  Reduce  f  and  f  to  ninths.  What  is  the  least  com- 
mon multiple  of  the  denominators  of  -§  and  -J  ? 

9.  What  is  the  least  common  denominator  of  §  and  f  ? 

10.  What  is  the  least  common  denominator  of  £,  \ 
and  -&•? 

188.  Principle. — The  least  common  denominator  of 
two  or  more  fractions  is  the  least  common  multiple  of  thevr 
denominators. 

WRITTEN  EXERCISES. 

189. — 1.  Reduce  f  and  f  to  fractions  having  a  com- 
mon denominator. 

3X2 6  Solution. — Since  2  times  the  number  of  fourths 

4  X  2  8  must  be  the  number  of  eighths,  multiply  both  terms 
of  |  by  2,  which  gives,  as  its  equivalent,  f.  Hence,  f  and  f  are 
the  fractions  required. 

2.  Reduce  J-,  f  and  fV  to  fractions  having  the  least 
common  denominator. 

1X12  _    12  Solution. — The  least  common  multiple  of  the 

2X12       24.        denominators  of  the  fractions  is  24.     Hence,  24 

SX3 9_       must  be  their  least  common  denominator. 

8^3       $4  Since  the  required  denominator  24  is  12  times 

5X~  — :  **L  the  denominator  of  \,  multiply  both  its  terms  by 
12X2  24  12^  reducing  it  to  ||.  Since  the  required  de- 
nominator is  3  times  the  denominator  of  f ,  multiply  both  its 
terms  by  3,  reducing  it  to  ^ ;  and  since  the  required  denomi- 
nator is  2  times  the  denominator  of  T\,  multiply  both  its  terms 
by  2,  reducing  it  to  ||.  Hence,  |f,  ■£%  and  ^J  are  the  fractions 
required. 

3.  Reduce  |  and  ^-  to  fractions  having  a  common 
denominator. 

4.  Reduce  i,  f  and  f  to  fractions  having  the  least 
common  denominator. 


FRACTIONS.  91 

190.  Rule  for  Reducing  Fractions  to  a  Common  Denominator.— 

Multiply  both  terms  of  each  fraction  by  any  num- 
ber that  will  mahe  their  denominators  alike. 

191.  Rule  for  Reducing  Fractions  to  the  Least  Common  De- 
nominator.— Find  the  least  common  multiple  of  all  the 
denominators  for  the  least  common  denominator, 
and  multiply  both  terms  of  each  fraction  by  such  a 
number  as  will  reduce  it  to  that  denominator. 

PROBLEMS. 

192. — 1.  Change  f  and  f  to  fractions  having  a  com- 
mon denominator. 

2.  Change  f ,  TV  and  ^  to  fractions  having  a  common 
denominator. 

3.  Change  yy,  ^  and  ^  each  to  eighty-eighths. 
Reduce  to  the  least  common  denominator — 


4. 

2 

4 
9 

and 

tV 

7. 

2 

T3o-  and 

1 1 

14* 

10. 

2 

f  and  f . 

5. 

1 

-2' 

5 

a 

and 

3 

S* 

8. 

1 

-&  and 

8 
21« 

11. 

2 
T3" 

,  yi  and 

If 

6. 

9 

5J 

7 
8 

and 

9 

io- 

9. 

4 

if 

I  and  |f. 

12. 

1  1 
16"' 

A  and 

21 
45' 

193.  Test  Questions. — 1.  When  is  a  fraction  reduced  to 
higher  terms?     When  to  lower  terms? 

2.  How  do  you  reduce  an  integer  or  a  mixed  number  to  an 
improper  fraction?  How  do  you  reduce  an  improper  fraction 
to  an  integer  or  a  mixed  number? 

3.  What  effect  upon  the  value  of  a  fraction  has  the  multiply- 
ing of  both  terms  by  the  same  number?  The  dividing  of  both 
terms  by  the  same  number? 

4.  How  do  you  reduce  fractions  to  higher  terms  ?  To  lower 
terms  ? 

5.  When  have  fractions  a  common  denominator?  What  is 
the  least  common  denominator  of  two  or  more  fractions?  How 
do  you  reduce  fractions  to  fractions  having  a  common  denomi- 
nator?    The  least  common  denominator? 


92  FM  ACTIONS. 

SECTION    XV. 

ADDITION  OF  FRACTIONS. 

194. — 1.  John  has  f  of  a  dollar,  and  his  brother  has 
f .     How  many  eighths  have  both  ? 

2.  If  Jane  has  f  of  an  orange,  and  Susan  has  £,  ho\y 
many  fifths  have  both  ? 

3.  How  many  ones  are  f  and  £  ?     {$  and  TV  ? 

4.  How  do  you  add  fractions  having  a  common  de- 
nominator ? 

By  adding  their  numerators  and  writing  their  sum  over  the 
denominator. 

5.  Jason  has  f  of  an  acre  planted  with  corn,  and  f  of 
an  acre  with  potatoes.     How  many  acres  has  he  planted  ? 

Solution. — He  has  planted  f  of  an  acre  +  f  of  an  acre,  f  is 
equal  to  r92,  and  f  to  T82 ;  and  T\  and  T82  are  }£,  or  lr52.  Hence, 
he  has  planted  1T52  acres. 

6.  How  can  fractions  which  have  different  denomi- 
nators be  added  ? 

By  first  reducing  them  to  a  common  denominator. 

7.  Mary  has  §  of  an  apple  and  Ella  has  f  of  an  apple. 
How  many  apples  have  both  ? 

8.  Henry  gave  -fa  of  a  dollar  for  a  book  and  f  of  a  dol- 
lar for  a  knife.    How  many  dollars  did  he  give  for  both  ? 

9.  How  many  are  £  and  f  ?     f  and  f? 

10.  A  man  having  undertaken  a  piece  of  work,  per- 
formed \  of  it  the  first  day,  \  of  it  the  second,  and  \  of 
it  the  third.  How  much  of  it  did  he  perform  in  the 
three  days  ? 

11.  How  many  are  f  and  $■?     f  and  f  ? 


FRACTIONS.  93 

12.  A  man  bought  1^  bushels  of  corn  at  one  time  and 
3f  bushels  at  another.  How  many  bushels  did  he  buy 
in  all  ? 

115,  Principle. — Fractions  must  express  like  parts,  or 
have  a  common  denominator,  before  they  can  be  added. 

WRITTEN  EXERCISES. 

196.— 1.  What  is  the  sum  of  f  f  and  f  ? 

£ _L *  i  !^-  U  =  a         Solution. — Since  the  fractions  all 

7       7       7  ■       7  express  like  parts,  their  sum  may  be 

found  by  adding  their  numerators.     5  sevenths  -f-  3  sevenths  + 
6  sevenths  =  V4>  or  2. 

2.  What  is  the  sum  of  ft,  if-  and  if? 

3.  What  is  the  sum  of  f^,  f  f  and  $$  ? 

4.  What  is  the  sum  of  ^,  f  and  f  ? 

Solution.  —  Reducing  the 
fractions  to  their  least  common 
denominator,  we  have  £$  +  f| 
+  H  =  tt,  °r  1H,  which  is  the 
sum  required. 

5.  What  is  the  sum  of  f ,  $  and  \\  ? 

6.  What  is  the  sum  of  f,  &  and  ft  ? 

7.  What  is  the  sum  of  3f,  5£  and  11  ? 

?            *  Solution. — Reducing  the  fractions 

^4  ~      ° 8  to  their   least   common   denominator 

r  1 ~4  and  adding  their  numerators,  we  have 

* "         8  as  their  sum  \°  =  If.     Write  the  § 

11    =11  under  the  fractions,  and   add  the  1 

«,,«,       ~9r7—  9D1  With  the  inteSers'  SivinS  2°t  =  2°i' 

awn,    ^  —  40-       the  sum  required> 


i_ 

«0 

f_ 

Af 

8  _ 

15 

g  ■* 

-  40'} 

J 

"^T 

s  ~~ 

z  40 

20 

*0 

.     32 

+ 

15 

40- 

67 

40 

8.  What  is  the  sum  of  11  J,  15|  and  5^ 


_9 

2   ' 


94  FRACTIONS. 

197.  Rule  for  Addition  of  Fractions.— Reduce  the  frac- 
tions to  equivalent  fractions  having  a  common  de- 
nominator; add  their  numerators ,  and  write  und  ,r 
the  sum  the  common  denominator. 

If  there  be  mixed  numbers  or  integers,  add  the 
fractions  and  integers  separately,  and  uniii  the 
results. 


PROBLEMS. 

8. 

What  is  the  sum 

of- 

1. 

2. 
3. 
4. 

MandH?     . 
i,  T3¥  and  tV  ? 

9      16.  „n A   11  9 
2  5'   2  5   dIlu   2  5  • 

Y%  ^andfV? 

5. 
6. 

7. 
8. 

|,  land  5? 
8,  T8Tand27V? 
31  4|  and  33? 
H.^andl^? 

9.  What  is  the  sum  of  -J  +  fV  +  I  +  H  ? 

10.  What  is  the  sum  of  3J  +  2£  +  f  +  $  ? 

11.  A  farmer  has  in  one  bin  31f  bushels  of  wheat,  in 
a  second  bin,  15f  bushels,  and  in  a  third,  16^  bushels. 
How  many  bushels  has  he  in  the  three  bins  ? 

12.  If  you  should  spend  6-^5-  hours  in  study,  3  2V 
hours  in  play,  and  11^  hours  in  sleep,  how  many  hours 
would  you  spend  in  all  ? 


SECTION   XVI. 

SUBTRACTION  OF  FRACTIONS. 

199. — 1.  John  and  his  brother  have  f  of  an  apple. 
If  John  has  f ,  what  part  of  the  apple  has  his  brother  ? 

2.  Jane  has  |  of  an  orange ;  if  she  should  give  Susan 
|,  what  part  of  the  orange  would  she  have  left  ? 


FRACTIONS.  95 

3.  How  many  fifths  are  |  less  f  ?     V  less  i  ? 

4.  How,   then,   do   you   subtract   one  fraction  from 
another  when  both  have  a  common  denominator  ? 

5.  If  f  of  an  apple  be  taken  from  it,  what  part  will 
remain  ? 

6.  How  much  is  1  less  \  ?     2  less  \  ?     3  less  f  ? 

7.  If  a  cord  of  wood  cost  f  of  a  dollar  less  than  6 
dollars,  what  is  its  cost  ? 

8.  How  much  is  \  less  -f-  ? 

•  Solution. — \  equals  T7¥,  and  }  equals  T4¥ ;  and  T7¥  less  T4¥  is  T\. 

9.  A  man  worked  T97  of  a  day,  and  his  son  f  of  a  day. 
How  much  longer  did  the  man  work  than  his  son  ? 

10.  How  much  is  \  less  \  ?     \  less  -§  ?     |-  less  f  ? 

11.  How  do  you  prepare  fractions  having  different 
denominators  for  subtraction  ? 

12.  If  a  man  owning  -ff  of  a  mill  should  sell  f  of 
the  mill,  what  part  of  it  would  he  have  left  ? 

13.  How  many  are  2£  less  f  ?     5f  less  2£? 

14.  What  number  must  be  added  to  3|  to  make  9  ? 

200.  Principle. — Fractions  must  express  like  parts,  or 
have  a  common  denominator,  before  they  can  be  subtracted. 

WRITTEN  EXERCISES. 

201.— 1.  What  is  the  difference  between  f  and  f  ? 

t    -9 6_ 2 6-2 4         Solution.  —  Reducing     the 

8  9~9  9~  9  ~9  fractions  to  their  least  com- 
mon denominator  and  finding  the  difference  between  their 
numerators,  we  have  f,  the  difference  required. 

2.  Find  the  difference  between  -fa  and  f . 

3.  Find  the  difference  between  T4ir  and  J. 


96  FRACTIONS. 

4.  Find  the  difference  between  8$  and  5|. 

Solution. — Eeduce    the    fractions    to 

ol  _ -  qA_  — -  y~        fractions  having  the  least  common  denomi- 

4  12  12        nator.     Since  A  cannot  be  taken  from  A, 

5~  ==  5 —  =  5~—       we  take  1  one,  or  }$,  from  the  8  ones, 

— —       leaving  7  ones,  and  adding  the  \\  to  the 

Difference,      2^       T32,  have  \\ ;  5T82  from  7^|  leaves  2TV,  the 

difference  required. 

5.  Find  the  difference  between  14$  and  6f . 

6.  Find  the  difference  between  28f-  and  9y\. 

202.  Rule  for  Subtraction  of  Fractions.— Reduce  the  frac- 
tions to  equivalent  fractions  having  a  common  de- 
nominator, and  write  the  difference  of  the  nume- 
rators over  the  common  denominator. 

If  there  be  mixed  numbers,  subtract  first  the 
fractional  part  of  the  subtrahend,  and  then  the 
integral  part,  and  unite  the  results. 


PROBLEMS. 

203.  Subtract— 


1    $f  from  f 

2.  A 
3 


4.  T\  from  $$. 

5.  A  from  T5r. 


7.  13$  from  16$. 

8.  llj  from  19}. 

9.  33$f  from  100. 


6.  f$  from  $f 

10.  If  a  farmer  should  sell  24^  acres  from  a  lot  con 
taining  66f  acres,  how  many  acres  would  be  left? 

11.  What  is  the  difference  between  85$  and  83$? 


204.  Test  Questions.— 1.  What  is  the  principle  in  addition 
of  fractions  ?  Plow  do  you  add  when  all  the  numbers  are  frac- 
tions ?   When  there  are  fractions  and  mixed  numbers  or  integers  ? 

2.  What  is  the  principle  in  subtraction  of  fractions?  How 
do  you  subtract  when  the  minuend  and  subtrahend  are  frac- 
tions?   When  the  minuend  and  subtrahend  are  mixed  numbers? 


FRACTIONS. 


97 


SECTION   XVII. 
MULTIPLICATION  OF  FRACTIONS. 

CASE   I. 

Fractions  Multiplied  by  Integers. 

205. — 1.  If  you  should  give  \  of  a  melon  to  each  of 
5  boys,  how  many  sixths  of  a  melon  would  they  all 
have? 

2.  If  you  should  give  ^  of  a  cake  to  each  of  7  boys, 
how  many  thirds  of  a 
cake  would  be  requir- 
ed? How  many  cakes  ? 

3.  7  times  \  are  how 
many  times  1  ? 

4.  If  1  yard  of 
cloth  cost  |  of  a  dol- 
lar, how  many  dollars 
will  8  yards  cost? 

Solution.  —  Since    1 
yard  of  cloth  cost  f  of  a 
dollar,  8  yards  must  cost  8  times  |  of  a  dollar,  which  are  V,  or 
y>,  equal  to  6f  dollars. 

5.  At  ■£$  of  a  dollar  each,  how  many  dollars  will  9 
hats  cost  ? 

6.  If  1  pound  of  tea  cost  f  of  a  dollar,  how  many 
dollars  will  7  pounds  cost  ? 

7.  7  times  £  are  how  many  times  1  ? 

8.  If  a  horse  eat  \\  of  a  bushel  of  grain  in  1  week, 
how  many  bushels  will  he  eat  in  10  weeks  ? 

9.  How  many  are  6  times  f  ?     7  times  f  ? 

9 


98  FRACTIONS. 

10.  How  many  are  8  times  yy  ?     5  times  yy  ? 

11.  If  ^  of  an  acre  will  pasture  a  cow,  how  many 
acres  will  pasture  12  cows? 

12.  How  many  are  11  times  |?     12  times  f  ? 

13.  If  f  of  a  yard  of  cloth  is  y  of  what  is  required 
for  a  suit  of  clothes,  how  many  yards  will  be  required 
for  the  suit  ? 

14.  If  you  can  earn  ^  of  a  dollar  in  1  day,  how 
many  dollars  can  you  earn  in  1 1  days  ? 

15.  Multiplying  the  numerator  of  ^  by  11  gives  ff, 
or  |;  dividing  the  denominator  of  ^  by  11  gives  f ; 
in  each  case  the  fraction  is  multiplied  by  11.  How, 
then,  may  a  fraction  be  multiplied  by  an  integer  ? 

16.  Multiply  Y5¥  by  8.     A  by  7.     yV  by  5. 

17.  How  many  are  5  times  6J? 

Solution. — 5  times  6  are  30,  and  5  times  \  are  },  or  2J,  which 
added  to  30  gives  32J,  the  answer  required. 

18.  If  1  ton  of  coal  cost  5f  dollars,  how  many  dollars 
will  7  tons  cost  ? 

19.  If  you  can  gather  6f  quarts  of  berries  in  1  day, 
how  many  quarts  can  you  gather  in  6  days  ? 

20.  At  the  rate  of  3|  miles  in  1  hour,  how  far  can 
you  walk  in  8  hours  ? 

21.  How  much  must  be  given  for  11  pounds  of  tea,  at 
If  dollars  a  pound  ? 

22.  What  will  5  yards  of  cloth  cost,  at  3f  dollars  a 
yard? 

206.  Principle. — Multiplying  the  numerator,  or  di- 
viding the  denominator,  by  any  number,  multiplies  the 
fraction  by  that  number. 


FRACTIONS. 


99 


WRITTEN   EXERCISES, 


207.— 1.  Multiply  H  by  8. 

1L  V  Q  -  UX8—  88  _r   8   __  r>l 

16   X*-~16~~  16  -°16  ~b2 


Or, 


n. 


11 


-  y  R  —  -  ±-  =  —  =  £- 

16  ^  16  +  8        2  2 


Solution.— 8  times  11 
sixteenths  are  f f,  which, 
reduced,  is  5t8q,  or  5|,  the 
result  required. 

Or,  since  dividing  the 


denominator  of  a  fraction  multiplies  the  fraction,  8  times  \\  = 
V,  or  5 J,  the  same  result. 

2.  Multiply  llf  by  7. 

4  4 


11X7  =  77 


jX7=f= 


834 


Solution.— Since  llf  equals  11-f-f,  the 
product  of  llf  by  7  is  the  same  as  7  times 
11  plus  7  times  f.  7  times  11  are  77,  and 
7  times  |  are  V,  or  b\.  77  +  5 J  =  82J, 
the  product  required. 


3.  Multiply  H  by  9.     ffo  by  17.     «HJ  by  15. 

4.  Find  the  product  of  25J  by  16. 

208.  Rules  for  Multiplying  a  Fraction  by  an  Integer.— Mul- 
tiply the  numerator,  or  divide  the  denominator,  by 
the  integer. 

If  the  multiplicand  be  a  mixed  number,  multi- 
ply the  integer  and  fraction  separately,  and  add 
the  products. 


PROBLEMS. 

209.  Multiply- 

1.  t^  bv  9. 

5.  tf  by  15. 

9. 

3fbyl7. 

2.  H  by  20. 

6.  if  by  9. 

10. 

1\  by  14. 

3.  i  by  25. 

7.  A  by  8. 

11. 

4^by31 

4.  &  by  13. 

8.  if  by  18. 

12. 

19  j  by  63 

100  FRACTIONS. 

CASE   II. 

Integers  Multiplied  by  Fractions. 

210. — 1.  \  of  3  inches  is  what  part  of  1  inch  ? 

Solution. — \  of  3  inches  must  be  3  times  \  of  1  inch,  or  | 
of  linen. 

2.  \  of  3  apples  is  how  many  apples  ? 

3.  i  of  8  bushels  is  how  many  bushels  ? 

4.  Arthur  has  5  dollars,  and  James  has  §  as  many. 
How  many  dollars  has  James  ? 

Solution. — James  has  f  of  5  dollars.  Since  J  of  5  dollars  is 
f  of  1  dollar,  §  of  5  dollars  must  be  2  times  f,  which  are  l°.  or 
3J  dollars.     Therefore  James  has  3J  dollars. 

5.  How  much  will  f  of  a  yard  of  cloth  cost,  at  the 
rate  of  7  dollars  a  yard  ? 

6.  f  of  7  is  what  number? 

7.  If  a  man  can  do  a  piece  of  work  in  45  days,  in 
how  many  days  can  he  do  f  of  it  ? 

8.  f  of  45  is  what  number  ? 

9.  If  a  barrel  of  flour  is  worth  16  dollars,  how  much 
is  ^  of  a  barrel  worth  ? 

10.  -f-  of  42  is  what  number? 

11.  Multiply  9  by  f.     5  by  f     8  by  f 

12.  At  6  dollars  a  ton,  what  will  5f  tons  of  coal  cost? 

Solution. — Since  1  ton  costs  6  dollars,  5f  tons  will  cost  5§ 
times  6  dollars ;  5  times  6  dollars  are  30  dollars,  and  f  of  6  dol- 
lar are  V,  or  4|  dollars;  30  dollars  and  4J  dollars  are  34| 
dollars,  the  cost  required. 

13.  At  5  dollars  a  yard,  what  will 
sost? 

14.  Multiply  8  by  4|.     7  by  3$. 


FRACTIONS.  101 

15.  If  6  men  can  do  a  piece  of  work  in  4|  days,  how 
many  men  will  it  take  to  do  it  in  1  day  ? 

16.  Multiply  4  by  &ft_.     8  by  8|.     5  by  7TV 

211.  Principle. — A  number  is  multiplied  by  a  fraction 
by  obtaining  such  a  part  of  the  number  as  the  fraction 
indicates. 

WRITTEN  EXERCISES. 

212.— 1.  Multiply  35  by  f,  or  find  f  of  35. 

Solution.— Since  f  ==  \  of  5, 
f  times  35  must  equal  \  of  5 
times  35,  or  357x  -,  which,  by  can- 
celing, or  §~I>  or  25. 

Or,  since  f  =  5  times  \t  we 
find  f  of  35  by  taking  5  times  \ 
of  35 ;  \  of  35  is  5,  and  5  times 
5  are  25. 

2.  Multiply  m  by  ^  or  find  T4T  of  66. 

3.  Multiply  75  by  ft  or  find  \\  of  75. 

213.  Rules  for  Multiplying  an  Integer  by  a  Fraction.— Mul- 
tiply the  integer  by  the  numerator  of  the  fraction, 
and  divide  the  product  by  the  denominator.     Or, 

Divide  the  integer  by  the  denominator  of  the 
fraction,  and  multiply  the  quotient  by  the  nume- 
rator. 


35x1  = 

-^ 

il  — 

25 

Or, 

35xf  = 

i 

5 
■35- 

'  9- 

\ 

X5 

=  25 

PROBLEMS. 

214.  Multiply— 


1.  72  by  A- 

2.  96  by  f . 

3.  105  by  -&. 


4.  112  by  f. 

5.  215  by  fV. 

6.  360  by  $. 


7.  327  by  A- 

8.  516  by  f. 

9.  -819  by  A- 


102 


FRACTIONS. 


CA.S1S   III. 

Fractions  Multiplied  by  Fractions. 

215. — 1.  If  ^  of  a  pear  be  separated  into  two  equal 
parts,  what    part    of 
the  pear  will  one  of 
those  parts  be  ? 

2.  J  of  \  is  what 
part  of  1  ? 

3.  If  |  of  an  orange 
be  equally  shared  by 
two  boys,  what  part 
of  the  orange  will 
each  receive? 

4.  \  of  |  is  what 
part  of  1  ? 

Solution.— \  of  f  is  equal  to  2  times  \  of  \  ; 
2  times  \  are  §,  or  J,  the  part  required. 

5.  What  is  §  of  f?     J  of  |?     i  of  |? 


i  of  i  is  k  and 


6.  What  is  *of  *?     lof  #? 


ioff? 


7.  A  man  owning  J  of  a  ship  sold  f  of  his  share. 
What  part  of  the  ship  did  he  sell  ? 

Solution.— He  sold  f  of  ]  of  the  ship ;  f  of  J  is  equal  to  3 
times  \  of  J ;  \  of  |  is  equal  to  ¥V,  and  3  times  ^  are  §  |. 
Therefore,  he  sold  JJ  of  the  ship. 

8.  When  a  man  had  traveled  y9^  of  a  mile  he  had 
still  to  travel  a  distance  equal  to  f  of  that  gone  over. 
How  far  had  he  still  to  travel,  and  how  far  did  he  travel 
in  all? 


9. 
10. 


What  is  f  of  4?     |  of  £? 


What  is  |  of  T2T? 


I  of*? 

A  of  I?    fof|? 


FRACTIONS.  103 

11.  What  will  f  of  a  bushel  of  corn  cost,  at  ^  of  a 
dollar  a  bushel  ? 

12.  At  2\  dollars  a  yard,  what  is  the  cost  of  f  of 
a  yard  of  cloth  ? 

Solution.— Since  1  yard  costs  2|  dollars,  |  of  a  yard 
will  cost  f  of  2|  dollars.  2-|  dollars  are  equal  to  |  of  a 
dollar ;  f  of  |  are  f  f ,  or  2TV  Hence,  £  of  a  yard  will  cost  2T11 
dollars. 

13.  If  a  cord  of  wood  cost  3£  dollars,  how  much  will 
f  of  a  cord  cost  ? 

14.  What  is  f  of  5i?    fof3^?    fof7i? 

15.  What  is  f  of  2i?    f.of  7£?    f  of  2£  ? 

16.  When  fractions  are  connected  by  the  word  of 
what  does  the  of  denote  ?  Multiplication. 

WRITTEN  EXERCISES. 

216.— 1.  Multiply  f  by  f,  or  find  f  of  f. 

0      &      a ,        0  Solution. — f  of  |  is  the  same  as  3 

o  v,   o  41/. 8^ 

9^7~63~~Tl  times  \  of  f ;  f  of  |  is  g^,  or  &,  and  3 

Or,  times  -$$  is  ~G1^,  or  §f,  which,  reduced, 

£      3      8X3-  __   8  is  28t»  tae  product  required.     Or,  indi- 

9      7  ~~  5-x  7  ~  21  eating  the  multiplication  and  cancel- 

3  ing,  we  have  ^8T,  as  before. 

2.  What  is  the  product  of  8|  by  4£  ? 

2     26         1      21  Solution. — Reducing    the    mixed 

S'3  =  J"" ;  4j  ~  J~*  numbers  to  equivalent  fractions,  we 

7  have  8|  X  4|  =  y  X  V-     Canceling 

^X^  =  1S2  =  36-  and  multiPlying>  we  have  if*,  which, 

*"       5        J            5  reduced,  is  36f,  the  product  required. 

3.  Multiply  iV  by  £-.  or  find  f  of  &. 

4.  Multiply  13f  by  2|.     29^  by  6£. 


104 


FRACTIONS. 


217.  Rules  for  Multiplying  a  Fraction  by  a  Fraction.— Multi- 
ply the  nuinerators  together  for  the  numerator,  and 
the  denominators  for  the  denominator  of  the  pro- 
duet. 

If  there  be  mixed  numbers,  reduce  them  to  frac- 
tions and  then  multiply. 


1. 
2. 
3. 


13.  AVhat  is  the  value  of  £  of  if  ? 

14.  What  is  the  value  of  f  of  \  of  f  f  ? 

15.  John  owns  f  of  a  boat,  and  his  brother  owns  -^  as 
much  of  it.    What  part  of  the  boat  does  his  brother  own  ? 

16.  What  will  \  of  a  yard  of  cloth  cost,  at  3^  dollars 
a  yard  ? 


PROBLEMS. 

218.  Multiply- 

&byf. 

5.  m  by  t5t- 

9. 

tt  by  7|, 

Abyf. 

6.  3^  by  I 

10. 

1*  by  ft. 

A  byf 

7.  16|  by  f. 

11. 

12  by  If. 

t2oV  by  f 

8.  U  by  3*. 

12. 

17|  by  5A- 

219.  Test  Questions.— 1.  In  what  two  ways  may  a  fraction 
be  multiplied?  Show  that  multiplying  the  numerator  multi- 
plies the  fraction.  That  dividing  the  denominator  multiplies 
the  fraction. 

2.  How  is  an  integer  multiplied  by  a  fraction?  Show  how 
you  can  take  of  a  number  the  part  denoted  by  a  fraction.  In 
what  two  ways  may  an  integer  be  multiplied  by  a  fraction  ? 

3.  How  do  you  multiply  a  fraction  by  a  fraction  ? '  How  do 
you  proceed  if  there  are  mixed  numbers? 

4.  When  fractions  are  connected  by  the  word  of,  what  does 
the  of  denote  ? 


FRACTIONS.  105 

SECTION    XVIII. 
DIVISION  OF  FRACTIONS. 

CASE   I. 

Fractions  Divided  by  Integers. 

220, — 1.  A  boy  having  f  of  an  orange  wished  to  di- 
vide it  equally  among  3  boys.  What  part  of  the  orange 
could  he  give  to  each  ? 

2.  \  of  f  is  what  part  of  1  ? 

3.  If  5  books  can  be  bought  for  f  of  a  dollar,  what 
will  1  book  cost? 

4.  If  4  yards  of  cloth  can  be  bought  for  £  of  a  dollar, 
what  will  1  yard  cost? 

Solution. — If  4  yards  of  cloth  cost  f  of  a  dollar,  1  yard  will 
cost  I  of  |  of  a  dollar,  or  ^  of  a  dollar. 

5.  If  1  man  can  do  a  piece  of  work  in  f  of  a  day,  in 
what  time  can  5  men  do  it  ? 

6.  i  of  |  is  what  part  of  1  ? 

7.  When  6  books  of  equal  value  cost  V2  of  a. dollar, 
what  is  the  cost  of  1  book  ? 

8.  I  divided  by  6,  or  |  of  f,  is  what  number? 

9.  Divide  i  by  2.     f  by  3.     f  by  5. 

10.  Divide  f  by  3.     £  by  4.     jf  by  5. 

11.  Dividing  the  numerator  of  \j  by  5  gives  -fa  ; 
multiplying  the  denominator  of  yy  by  5  gives  £§-,  which 

•equals  fa ;   in  each  case  the  fraction  is  divided  by  5. 
How,  then,  may  a  fraction  be  divided  by  an  integer  ? 

12.  Divide  V6  by  8.     f  by  6.     if  by  7. 

13.  Divide  M  by  3.     \°  by  5.     V8  by  9. 


106  FRA  cnoxs. 

14.  If  1  man  can  do  a  piece  of  work  in  8|  days,  in 
what  time  can  3  men  do  it  ? 

Solution. — If  1  man  can  do  a  piece  of  work  in  8|-  days,  3 
men  can  do  it  in  J  of  8|  days,  or  in  2\\  days. 

15.  If  4  bushels  of  corn  cost  5|  dollars,  what  is  the 
cost  of  1  bushel  ? 

16.  Divide  12J  by  7.     9|  by  6.     10}  by  9. 

221.  Principle. — Dividing  the  numerator  or  multiplying 
the  denominator  by  any  number  divides  the  fraction  by 
that  number. 

WRITTEN  EXERCISES. 

222.— 1.  What  is  the  quotient  of  jf  divided  by  6  ? 

12  12^-6       2  Solution. — Since  dividing  the 

Js  ^  ls~  ~  13        numerator  of  a  fraction  divides  the 

fraction,  \\  divided  by  6  gives  T23, 
2  the  quotient  required. 

19  -19  2 

u '.'^  6  —  m~~^-— 'if  Or>   since   multiplying    the    de- 

nominator of  a  fraction  divides  the 
fraction,  Jf  divided  by  6  =  Jzxe  ~  "h*  tne  same  result. 

2.  What  is  the  quotient  of  31^  divided  by  5  ? 

«5  Solution.— Since  31  \  is  ±f£, 

-_®L  =  (}1       we  may  divide  31 J  by  5  by  di- 
viding its  equivalent,  *£*,  which 
gives  2j5,  or  6},  the  quotient  re- 
£  quired. 

/?A  Or,  we  may  divide  the  mixed 

^  number  without  first  reducing  it 

to  an  improper  fraction.  One  fifth  of  31  is  6,  with  a  remainder 
1,  which  is  equal  to  f;  this  added  to  the  J  gives  {.  One 
fifth  of  J  is  I,  which  added  to  6  gives  6\,  the  same  result. 

3.  Divide  &  by  3.     &  by  34.     23  j  by  7. 


°r>      i      i 
5)3lh 


FRACTIONS. 


107 


223.  Rules  for  the  Division  of  a  Fraction  by  an  Integer.— 
Divide  the  numerator,  or  multiply  the  denomin- 
ator, by  the  integer. 

If  the  dividend  is  a  mixed  number,  reduee  it  to 
an  improper  fraction  before  dividing ;  or,  divide 
the  integer  and  fraction  separately ,  and  unite  the 
results. 

PROBLEMS. 


224.  Divide— 


1.  I  by  6. 

2.  |  by  9. 

3.  H  by  22. 

10.  If   11 


4.  f  by  27. 


If  by  15. 


5. 

6.  H  by  7. 
boys  should   have  90|f  dollars  divided 


7.  16*  by  7. 

8.  18f  by  8. 

9.  25|  by  20. 


equally  among  them,  how  much  would  each  receive  ? 

CASE  II. 

Integers  Divided  by  Fractions. 

225. — 1.  How  many  thirds  of  a  cake  in  1  cake?  In 
2  cakes  ? 

2.  How  many  times 
f  of  a  cake  in  2  cakes  ? 

3.  How  many  pears, 
at  |  of  a  cent  each,  can 
be  bought  for  2  cents  ? 

4.  When  tea  is  f  of 
a  dollar  a  pound,  how 
many  pounds  can  be 
bought  for  6  dollars  ? 

Solution. — If  $  of  a  dollar  will  purchase  1  pound  of  tea,  6 
dollars  will  purchase  as  many  pounds  as  |  of  a  dollar  is  con- 
tained times  in  6  dollars.  6  dollars  equal  3T°  of  a  dollar ;  37°  -*-  f 
=  30  -s-  4  =  7|  oi  7£.     Hence,  7|  pounds  can  be  bought. 


108 


FRACTIONS. 


5.  At  |  of  a  dollar  a  yard,  how  many  yards  of  cloth 
can  you  buy  for  3  dollars  ? 

6.  1  is  how  many  times  f  ?     -f-? 

7.  2  are  how  many  times  f ' 


v 


I         WX    ^        AAW     If  J.AA14.XA      T  VAIAAVU         ^1         • 

i  are  how  many  times  f  ? 


t? 

1? 


i*-i=f+5=*o 


Or 


15^1     ^i  ^20 


WRITTEN  EXERCISES. 

226. — 1.  What  is  the  quotient  of  15  divided  by  f  ? 

Solution. — 15  is  equal  to  6¥°. 
60  fourths  divided  by  3  fourths 
gives  20,  the  quotient  required. 

Or,  since  15  -f- 1  is  15,  15  -=-  \ 
must  be  4  times  15,  and  15  -+- 1 
must  be  J  of  4  times  15,  which 
is  20,  the  same  result. 

2.  What  is  the  quotient  of  64  divided  by  $? 

3.  Divide  25  by  f .     112  by  |.     98  by  |. 

227.  Rules  for  Division  of  an  Integer  by  a  Fraction  —Rednei 
the  integer  to  a  fraction  having  the  same  denomi- 
nator as  the  divisor,  and  divide  the  numerator  of 
the  dividend  by  the  numerator  of  the  divisor.    Or. 

Multiply  the  integer  by  the  denominator  of  the 
divisor,  and  divide  the  result  by  the  numerator. 


228.  Divide— 

1.  15  by  |. 

2.  61  by  | . 

3.  21  by  f. 

4.  40  by  | 


PROBLEMS. 


9. 
10. 
11. 


24  by  f 
17  by 
36  by  ^ 


5.  51  by  &. 

6.  43  by  f 

7.  65  by  A- 

8.  90  by  A. 
13.  What  is  the  quotient  of  40  divided  by  3^? 

Solution. 
a=f,  and  jQ-m  320+25 


A- 


12.  110  by  f£. 


12**=  12* 


FRACTIONS.  109 

14.  Divide  17  by  2f.     28  by  If     42  by  6§. 

15.  At  -J  of  a  dollar  a  yard,  how  many  yards  of  cloth 
can  be  purchased  for  8f  dollars  ? 

CASE    III. 

Fractions  Divided  by  Fractions. 

229. — 1.  At  \  of  a  dollar  a  yard,  how  much  cloth  can 
be  bought  for  £  of  a  dollar  ? 

2.  At  ^  of  a  dollar  a  bushel,  how  many  bushels  of 
apples  can  be  bought  for  f  of  a  dollar  ? 

3.  How  many  times  is  f  contained  in  f  ? 

Solution. — f-^-f  is  equivalent  to  |$  divided  by  ^,  and  8 
twentieths  are  contained  in  15  twentieths  1J  times,  which  is  the 
result  required. 

4.  Divide  i  by  f .     §  by  f .     $  by  f     f  by  |. 

5.  At  jq  of  a  dollar  a  pound,  how  many  pounds  of 
sugar  can  be  bought  for  %  of  a  dollar? 

6.  At  j-Q  of  a  dollar  each,  how  many  books  can  be 
bought  for  2|  dollars  ? 

Solution. — 2f  is  equal  to  \ % ;  at  T3o  of  a  dollar  each,  as 
many  books  can  be  bought  for  \ %  of  a  dollar  as  3  tenths  are  con- 
tained times  in  24  tenths,  or  8,  which  is  the  number  required. 

7.  When  butter  is  \  of  a  dollar  a  pound,  how  many 
pounds  can  you  buy  for  2|  dollars  ? 

DEFINITION. 

230.  A  Complex  Fraction  is  one  that  has  a  fraction  in 
one  or  both  of  its  terms. 

3 

Thus,  -  is  a  complex  fraction,  and  indicates  the  division  of 

I  by  f. 

in 


110 


FRACTIONS. 


WRITTEN   EXERCISES. 

231. — 1.   What  is  the  quotient  of  f  divided  by  f  ? 

A  _j_  I  —  *t  _=_  ^  —  *?  -a  J?  —  #£         Solution.— f  and   § 

5      5      ^5       45       10        5         5      reduced  to  fractions  hav- 

ing  a  common  denominator  are  f  f  and  |§ ;  we  can  then  divide 

i  by  f,  by  finding  the  quotient  of  36  forty-fifths  divided  by  10 

forty-fifths,  which  is  3f ,  the  quotient  required.     Or, 


.tX9_18_ 


8        Since  f -*-l  is  f,  $  -4-  \  must  be 
5      .9       5X#       5       ^5     9  times  |,  or  ^~- ,  and  f  -*-  f  must 
be  ^  of  9  times  f ,  or  -f-^-f  .  which,  reduced,  is  3f ,  the  same  result. 

2.  What  is  the  quotient  of  ||  divided  by  f  ? 

3.  What  is  the  quotient  of  I  divided  by  21  ? 

232.  Rules  for  Division  of  a  Fraction  by  a   Fraction.— ife- 

£Zw,c£  the  fractions,  if  necessary ,  to  a  common  de- 
nominator, and  divide  the  numerator  of  the  divi- 
dend by  the  numerator  of  the  divisor.     Or, 

Multiply  the  dividend  by  the  denominator  of  the 
divisor,  and  divide  the  result  by  the  numerator. 


233.  Divid 

1.  f  by  &. 

2.  H  by  f . 

3.  H  by  f . 

4 


tfby*. 


PROBLEMS. 


5.  f  byf 


6-  A  by  |. 
7.  f  by  |. 
8-  H  by  5V 

13.  What  is  the  quotient  of  3£  divided  by  1£? 
Solution. 


9.  A  by  A- 
10.  A  by  &. 


11.  ,&bj  J. 

12.  &bj  &• 


2>i       25  ,     -,1       5 

?--y,and  7-^J; 


5 


-=^ 


REVIEW.  Ill 

14.  How  many  barrels  of  potatoes,  at  2|  dollars  a 
barrel,  can  be  bought  for  16^  dollars? 

15.  How  many  yards  of  calico,  at  T3g   of  a  dollar  a 
yard,  can  be  bought  lor  %  of  a  dollar  ? 

16.  What  is  the  value  of  A  ?     Of  2f  ? 

16  3 

17.  How  many  tons  of  coal,  at  3f  dollars  a  ton,  can 
be  bought  for  13 -J-  dollars  ? 


234:.  Test  Questions. — 1.  In  what  two  ways  can  a  fraction 
be  divided  by  an  integer?  If  the  dividend  be  a  mixed  number, 
how  do  you  divide  ? 

2.  How  do  you  divide  a  fraction  by  an  integer?  An  integer 
by  a  fraction  ? 

3.  What  is  a  complex  fraction?  Express  in  the  fractional 
form,  the  division  of  some  fraction  by  a  fraction.  How  do  you 
divide  a  fraction  by  a  fraction? 


SECTION   XIX. 
REVIEW  OF  FRACTIONS. 

235—1.  How  many  fifths  in  4|?     In  7f  ?     In  9|? 

2.  How  many  ones  in  V  ?     In  V  ?     In  V  ? 

3.  What  is  the  value  of  V  ?     Of  V  ? 

4.  What  improper  fraction  is  equal  to  6|  ? 

5.  Express  f  in  its  lowest  terms. 

6.  Express  f  in  higher  terms. 

7.  Reduce  f  and  f  each  to  eighteenths. 

8.  Reduce  f  and  |  to  fractions  having  a  common  de- 
nominator. 


112  BE  VIEW. 

9.  A  sum  of  money,  diminished  by  f  of  a  dollar,  is 
equal  to  227o  dollars.     What  is  the  sum  ? 

10.  If  I  sell  an  article  for  f  of  its  cost,  what  frac- 
tional part  of  the  cost  do  I  lose  ? 

11.  If  I  sell  an  article  for  J  part  more  than  its  cost, 
for  how  many  fourths  of  the  cost  do  I  sell  it  ? 

12.  If  I  pay  37^  cents  for  a  knife,  and  sell  it  for  |  of 
its  cost,  how  many  cents  do  I  lose  ? 

13.  A  fanner  had  ll-£  bushels  of  wheat  stolen,  which 
was  |  of  all  he  had.     How  much  had  he  ? 

14.  11^  is  ^  of  what  number? 

15.  If  a  slate  cost  ^V  of  a  dollar,  how  many  slates 
can  be  bought  for  If  dollars  ? 

WllTTTEX  EXERCISES. 

236.— 1.  Reduce  105f  to  eighths. 

2.  Reduce  j\  and  -^  to  fractions  having  the  least 
common  denominator. 

3.  Reduce  |-f  and  TV§-  each  to  its  lowest  terms. 

4.  Having  lost  f  of  a  dollar,  I  find  I  have  left  13f 
dollars.     What  sum  had  I  at  first  ? 

5.  A  merchant  owned  ff  of  a  ship,  and  sold  f  of  the 
ship.     What  part  of  the  ship  had  he  left? 

6.  5^  is  \  of  what  number  ?     ^  of  what  number  ? 

7.  if  is  \  of  what  number?     \  of  what  number? 

8.  \i  is  i  of  what  number?     TV  of  what  number? 

9.  If  a  horse  will  eat,  in  a  given  time,  ^i  of  a  ton  of 
hay ;  a  cow,  f  of  a  ton ;  and  an  ox,  t9q  of  a  ton,  what 
quantity  will  they  all  eat  in  the  same  time? 

10.  A  lady  gave  lOf  dollars  for  a  dress,  4^-  dollars 
for  a  shawl,  and  I  of  a  dollar  for  a  handkerchief.  How 
much  did  they  all  cost  her  ? 


REVIEW.  113 

11.  James  gave  -^  of  his  money  for  a  coat,  and  y2  °f 
it  for  a  library.     What  part  of  it  had  he  left  ? 

12.  What  will  19  pounds  of  tea  cost,  at  If  dollars  a 
pound  ? 

lo.  The  multiplicand  is  12|  and  the  multiplier  3 J. 
What  is  the  product  ? 

14.  Smith  owns  f  of  3^  of  a  ship,  and  Collins  owns 
f  as  much  as  Smith.     What  part  does  Collins  own  ? 

MENTA  L    EX  EH  CIS  ES. 

237.— 1.  20  is  f-  of  what  number? 

Solution. — If  20  is  f  of  some  number,  \  of  that  number 
must  be  \  of  20,  or  10,  and  f,  or  the  whole,  of  the  number, 
must  be  7  times  10,  which  is  70,  the  number  required. 

2.  40  is  f  of  what  number  ?     §  of  what  number  ? 

3.  30  is  f  of  what  number  ?     f  of  what  number  ? 

4.  16  is  f  of  what  number?     f  of  what  number? 

5.  20  is  f  of  what  number  ?     \  of  what  number  ? 

6.  If  f  of  an  acre  of  land  be  worth  40  dollars,  what 
is  an  acre  worth  ? 

7.  A  farmer  sold  a  cow  at  a  gain  of  20  dollars,  which 
was  y  of  her  cost.      What  was  the  cost  ? 

8.  Jane  is  16  years  old,  and  is  %  as  old  as  her  sister. 
What  is  the  age  of  her  sister  ? 

9.  A  pole  stands  y  of  its  length  in  the  mud,  \  in  the 
wrater,  and  the  remainder,  which  is  5  feet,  above  water. 
What  is  the  length  of  the  pole  ? 

10.  When  f  of  a  dollar  is  f  of  the  price  of  a  pound 
of  tea,  what  is  the  price  of  a  pound  ? 

Solution. — If  |  of  a  dollar  is  §  of  the  price  o£  a  pound,  \  of 
the  price  must  be  \  of  f,  or  |,  of  a  dollar,  and  f,  or  the  whole 
price,  must  be  3  times  f,  or  J,  equal  to  \\  dollars. 

111  * 


114  REVIEW. 

11.  f  is  |  of  what  number?     f  of  what  number? 

12.  If  |  of  a  rod  is  £  of  the  width  of  a  walk,  what  is 
the  width  of  the  walk  ? 

13.  f  is  f  of  what  number?     f  of  what  number? 

14.  How  many  times  f  is  \  ?     How  many  times  f-  is  f  ? 

15.  If  3  men  can  do  a  piece  of  work  in  4^  days,  in 
how  many  days  can  5  men  do  it  ? 

16.  A\  is  f  of  what  number?     f  of  what  number? 

17.  How  many  yards  of  cloth,  3  quarters  of  a  yard 
wide,  are  equal  to  7  yards  5  quarters  wide  ? 

18.  7  is  f  of  what  number?     f  of  what  number? 

19.  12  is  f  of  how  many  times  2  ? 

20.  30  is  |  of  how  many  times  12  ? 

21.  f  of  10  is  §  of  what  number? 

22.  f  of  12  is  f  of  how  many  times  4  ? 

23.  If  |  of  a  barrel  of  beef  cost  16  dollars,  how 
many  cords  of  wood  at  5  dollars  a  cord  will  pay  for  a 
barrel  of  beef? 

24.  If  f  of  a  ton  of  coal  cost  4  dollars,  what  will  | 
of  a  ton  cost  ? 

25.  If  f  of  a  yard  of  cloth  cost  3  dollars,  what  will 
f  of  a  yard  cost  ? 

WRITTEN  EXERCISES. 

238. — 1.  At  |  of  a  dollar  a  yard,  how  many  yards  of 
cloth  can  be  bought  for  16^  dollars? 

2.  Divide  W  by  18.     H  by  32.     3£  by  f. 

3.  I  paid  95^  dollars  for  flour,  at  the  rate  of  8^ 
dollars  per  barrel.     How  many  barrels  did  I  buy  ? 

4.  If  £  of  an  acre  of  land  cost  96  dollars,  what  will 
|  of  an  acre  cost  ? 


REVIEW.  115 

5.  If  the  divisor  is  y  and  the  quotient  5f,  what  is  the 
dividend  ? 

6.  If  |  of  a  ton  of  hay  cost  18  dollars,  what  will  Jf 
of  a  ton  cost  ? 

7.  If  the  product  of  two  factors  is  15f,  and  one  of 
the  factors  is  3^,  what  is  the  other  factor  ? 

8.  If  the  divisor  is  ^  and  the  dividend  is  -gfa,  what 
is  the  quotient  ? 

9.  I  own  \%  of  a  ship,  and  my  brother  owns  ^. 
How  many  times  as  much  as  he,  do  I  own  ? 

10.  John  sold  320  acres  of  land  ;  he  then  bought  f  as 
many  as  he  sold,  and  found  that  number  to  be  f  as 
many  as  he  had  at  first.     How  many  had  he  at  first  ? 


239.  Test  Questions.— 1.  What  is  an  integer  ?  A  fraction? 
A  mixed  number? 

2.  What  are  the  terms  of  a  fraction?  Which  term  is  the 
numerator?  Which  the  denominator?  How  is  a  fraction  ex- 
pressed by  figures  ? 

3.  What  is  a  proper  fraction?  An  improper  fraction?  A 
complex  fraction  ? 

4.  What  is  the  value  of  a  fraction?  Is  the  value  of  a  fraction 
changed  by  multiplying  both  terms  by  the  same  number?  By 
dividing  both  terms  by  the  same  number? 

5.  What  is  reduction  of  fractions?  How  may  fractions  be 
changed  to  fractions  with  higher  terms?  To  fractions  having 
lower  terms  ? 

6.  By  what  means  may  fractions  be  added  or  subtracted? 
How  are  fractions  with  different  denominators  prepared  for 
addition  or  subtraction? 

7.  What  is  a  principle  in  addition  of  fractions?  In  subtrac- 
tion of  fractions?  In  multiplication  of  fractions  ?  In  division 
of  fractions? 


116 


UNITED  STATES  MONEY. 


SECTION    XX. 

NOTATION   OF 

UNITED  STATES 

MONET. 

240.  United  States  Money  ^^^^ 

s  reckoned  in  dollars,  cents  and  mills.    %?  &  ^  &    i 

,WCENT/ 
10  mills  (m.)  are  1  cent. .  .c.  or  et. 

10  cents  "    1  dime d. 

10  dimes  "    1  dollar $. 

$l=10d~100ct,=1000m. 

241.  United    States    Money   consists   of    Coins   and 
Paper  Money. 

242.  Coins  are  pieces  of  metal   stamped  for  use  as 

noney. 

The  coins  of  the  United   States  are  now  made  of  gold,  silver, 
nckel  or  bronze. 


TABLE. 


UNITED    STATES   MONEY. 


117 


TABLE  OF  COINS. 

Names.  Values. 

Double-eagle....$20. 

Eagle 10. 

Half-eagle 5. 

3-dollar  piece...     3. 
Quarter-eagle...     21. 

Dollar 1. 

Bronze.  \  Cent lc. 


Gold. 


Silver. 


Nickel. 


Names.  Values. 

Dollar 100c. 

Half-dollar 50c. 

Quarter-dollar...  25c. 

20-cent  piece ....  20c. 
„  10-cent  piece....  10c. 

5-cent  piece 5c. 

[  3-cent  piece 3c. 


In  addition  to  the  above,  a  Trade-Dollar  is  issued  for  the  con- 
venience of  foreign  trade. 

243.  Paper  Money  consists  of  Notes,  issued  by  banks 
and  by  the  Treasury  of  the  United  States,  as  substi- 
tutes for  coin.  When  Treasury  Notes  are  of  less  face- 
value  than  $1,  they  are  called  Fractional  Currency. 

244.  In  business  transactions  dimes  are  regarded  as  a 
number  of  cents. 

Thus,  5  dimes  are  regarded  as  50  cents. 

In  denoting  by  figures  sums  of  United  States  money, 
the  sign  $  is  placed  before  dollars,  and  the  decimal  point 
(.)  is  placed  before  cents. 

Thus,  $5.90  expresses  five  dollars  and  ninety  cents. 
$.075  expresses  seven  cents  and  five  mills. 

245. — 1.  How  many  mills  in  1  cent  ?  In  3  cents  ?  In 
9  cents  ? 

2.  How  many  cents  in  10  mills  ?  In  30  mills  ?  In 
90  mills? 

3.  How  many  cents  in  1  dime  ?  In  5  dimes  ?  In  8 
dimes  ? 

4.  How  many  dimes  in  10  cents?  In  50  cents?  In 
80  cents  ? 


L18 


UNITED   STATES    MONEY. 


5.  How  many  dimes  in  1  dollar  ?     In  5  dollars  ? 

6.  How  many  dollars  in  10  dimes?     In  50  dimes? 

7.  How  many  cents  in  10  dimes  ?     In  1  dollar  ?     Ir 
I  dollars  ?     In  5  dollars  ? 

8.  How  many  dimes,  and  how  many  cents  over,  in  25 
;ents  ?     In  42  cents  ?     In  85  cents  ? 

9.  How  many  dollars,  and  how  many  cents  over,  in 
L25  cents  ?     In  250  cents  ?     In  375  cents  ? 

10.  How  many  cents,  and  how  many  mills  over,  in  15 
nills  ?     In  45  mills  ?     In  67  mills  ? 

11.  Since  10  mills  are  1  cent,  what  part  of  a  cent  is  1 
nill?     Is  2  mills?     Is  7  mills? 

12.  Since  10  cents  are  1  dime,  what  part  of  a  dime  is 
L  cent  ?     Is  3  cents  ?     Is  9  cents  ? 

13.  Since  10  dimes  are  1  dollar,  what  part  of  a  dol- 
ar  is  1  dime  ?     Is  2  dimes  ?     Is  7  dimes  ? 

14.  Since  100  cents  are  1  dollar,  what  part  of  a  dol- 
ar  is  1  cent?     Is  3  cents?     Is  17  cents? 

15.  Since  1000  mills  are  1  dollar,  what  part  of  a  dol- 
ar  is  1  mill  ?     Is  7  mills  ?     Is  117  mills  ? 


WRITTEN  EXERCISES. 

246.  Copy  and  read — 

1.  $15.25.      4.  $143.41.  7.  $3,057. 

2.  $7.08.        5.  $205.75.  8.  $1,305. 

3.  $97,375.    6.  $97,334.  9.  $41,065. 

247.  Write  in  figures — 

1.  Ten  dollars  sixteen  cents. 

2.  Twenty-seven  dollars  five  cents. 

3.  Sixty-three  dollars  thirty-one  cents. 

4.  Five  hundred  seventeen  dollars. 


10.  $16.15. 

11.  $11.31. 

12.  $97,005, 


UNITED  STATES  MONEY.  119 

SECTION   XXI. 

REDUCTION  OF  UNITED  STATES  MONEY, 

248. — 1.  Change  $7  to  cents  and  to  mills. 

7       =  No.  of  dollars.         Solution.— Since  $1  is  equal  to 
100  100  cents,  $7  must  equal  7  times  100 

YOO  =  No.  of  cents.        cents>  or  700  cents- 

1n  Since  1  cent  is  equal  to  10  mills, 

700  cents  must  equal  700  times  10 


7000  =  No.  of  mills.         mills,  or  7000  mills. 

2.  Change  $15  to  cents  and  to  mills. 

3.  Change  $15.67  to  cents. 

$15.67  =  Solution. — $15.67  may  be  changed  to  cents 

1567  cents       *>y  removing  the    dollar-sign  and    decimal 
point,  which  gives  1567  cents,  because  $15  = 
1500  cents,  and  1500  cents  plus  67  cents  are  1567  cents. 

4.  Change  $37.03  to  cents. 

5.  Change  $43,444  to  mills. 

$43,444=  Solution:— $43,444  may  be  changed  to 

A3  AAA  mills.       miUs  by  removing  the  dollar-sign  and  deci- 
mal point,  which  gives  43444  mills ;  because 
$43  =  43000  mills,  44  cents  =  440  mills,  and  43000  mills,  plus  440 
mills,  plus  4  mills,  are  43444  mills. 

6.  Change  $6,305  to  mills. 

7.  Change  7000  mills  to  cents  and  to  dollars. 

Solution. — Since  there  must  be 

7000  =  No.  of  mills.         one  ten^h  as  many  cents  as  there  are 

>vnn      xt      r        x  mills,  7000  mills  mav  be  changed  to 

700=  No.  of  cents.         centg  by  dividing  by  *10)  or  by  remov. 

7  =  No.  of  dollars,      ing  one  cipher  from  the  right,  which 
gives  700  as  the  number  of  cents. 


120  UNITED  STATES  MONET. 

Since  there  must  be  one  hundredth  as  many  dollars  as  there 
are  cents,  700  cents  may  be  changed  to  dollars  by  dividing  by 
100,  or  by  removing  two  ciphers  from  the  right,  which  gives  7 
a*  the  number  of  dollars. 

8.  Change  93000  mills  to  cents  and  to  dollars. 

249.  Rules  for  Reduction  of  United  States  Money.— lb  re- 
duce dollars  to  cents,  remove  the  dollar-sign  and 
annex  two  ciphers ;  to  reduce  dollars  to  mills,  an- 
nex three  ciphers ;  to  reduce  cents  to  mills,  annex 
one  cipher. 

To  reduce  dollars  and  cents  to  cents,  or  dollars, 
cents  and  mills  to  mills,  remove  the  dollar- sign 
and  the  decimal  point. 

To  reduce  cents  to  dollars,  point  off  two  orders 
from  the  right  and  prefix  the  dollar-sign /  to  re- 
duce mills  to  dollars,  point  off  three  orders  from 
the  right  and  prefix  the  dollar-sign;  and  to  re- 
duce mills  to  cents,  point  off  one  order  from  the 
right  and  prefix  the  dollar-sign. 

PROBLEMS. 

250.  Reduce— 

4.  5700  cents  to  dollars. 


1.  $57  to  cents. 

2.  $53  to  mills. 

3.  98  cents  to  mills. 


5.  53000  mills  to  dollars. 

6.  980  mills  to  cents. 


251.  Test  Questions.— 1.  In  what  is  United  States  money 
reckoned  ?    Of  what  does  it  consist  ?    Recite  the  table. 

2.  What  are  coins  ?  Of  what  are  the  coins  of  United  States 
money  made?  Name  the  coins  made  of  gold.  Of  silver.  Of 
nickel.  Of  bronze.  What  is  Paper  Money?  What  is  Frac- 
tional Currency  ? 

3.  How  are  dimes  regarded  in  business?  In  denoting  sums 
of  money  by  figures,  what  sign  denotes  dollars?  What  point 
is  placed  before  cents? 


V  NIT  ED  STATES  MONEY.  121 


SECTION    XXII. 

COMPUTATIONS  IN   UNITED  STATES 
MONEY. 

ADDITION. 

252. — 1.  If  you  should  pay  30  cents  for  a  slate,  25 
cents  for  a  writing-book  and  10  cents  for  a  pencil,  how 
much  would  you  pay  for  all  ? 

2.  How  much  is  30  cents  +  25  cents  +  10  cents  ? 

3.  Susan  gave  50  cents  for  a  collar,  40  cents  for  a 
thimble  and  12  cents  for  some  needles.  How  much  did 
she  give  for  the  whole  ? 

4.  If  you  give  $9.50  for  a  coat  and  $3.25  for  a  vest, 
how  much  do  you  give  for  both  ? 

WRITTEN  EXERCISES. 

253.— 1.  What  is  the  sum  of  $95.60,  $19  and  $4,375? 

$95.60  Solution. —Write  the  numbers  and  add,  as 

19.  required  by  the  rule  for  addition,  and  separate 

4-375      the  dollars  from  the  cents  in  the  sum  by  a  deci- 


118.975 

mal 

point. 

Write  and  add- 

(2.) 

(3.) 

(4.) 

(5.) 

$4-5.13 

$2,375 

$11.14 

$3.72 

5.07 

6.25 

63.15 

■  144 

17. 

7.625 

7.99 

5.138 

$67.20 

$9,002 

6.  $144.56  +  $17.18  +  $100.63  =-  what  amount? 


121  UNITEP  STATES  MONET. 

7.  Johnson  paid  for  a  farm  $6500,  and  for  improve- 
ments on  it  $365.50.  He  then  sold  it  for  $150  more 
than  the  whole  cost ;  for  how  much  did  he  sell  it  ? 

SUBTRACTION. 

254. — 1.  Arthur  had  95  cents,  and  gave  50  cents  for 
a  knife.     How  much  had  he  left  ? 

2.  85  cents  less  17  cents  are  how  many  cents? 

3.  If  you  had  $1.25,  and  should  give  75  cents  for  an 
arithmetic,  how  much  would  you  have  left? 

4.  What  sum  added  to  75  cents  will  make  $1.25? 
What  sum  added  to  50  cents  will  make  $1.25  ? 

5.  If  I  have  $9.25,  how  much  more  must  I  get  to 
pay  for  a  coat  worth  $10  ? 

6.  I  have  7  dimes;  how  much  more  must  I  get  to 
pay  for  a  book  worth  $1.10? 

WRITTEN  EXERCISES. 

255. — 1.  What  is  the  difference  between  $106  and 
$43.50? 

tQQ  Solution. — Write  the  numbers  and  subtract,  as 

AS  50  squired  by  the  rule  for  subtraction,  and  separate 

— -r '—  the  dollars  from  the  cents  in  the  difference  by  a 

$62.50  decimal  point. 

Write  and  subtract — 

(2.)  (3.)  (4.)  (5.) 

$164-15       $115,000       $31.15      $347.00 
87.09  37.085  4.17        243.19 

6.  How  much  less  is  $1867.25  than  $5555.43? 

7.  If  you  purchase  goods  at  a  cost  of  $316.50,  and 
Bell  them  for  $400,  how  much  do  you  gain? 


UNITED  STATES  MONEY. 


MULTIPLICATION. 


123 


56. — 1.  At    12   cents   each,   what  will   12  writing- 
books  cost  ? 

2.  At  25  cents  each,  what  will  5  spelling-books  cost  ? 

3.  James  paid  $1.25  for  a  hat,  and  Edward  paid  3 
times  as  much.     How  much  did  Edward  pay  ? 

4.  If  flour  is  $10  a  barrel,  what  will  10  barrels  cost? 

5.  What  will  5  bushels  of  wheat  cost,  at  $2.10  a 
bushel  ? 

WRITTEN   EXERCISES. 

257,-1.  What  is  the  product  of  $109.50  multiplied 
by  9? 

$109.50  Solution. — Write  the  numbers  and  multiply,  as 

q  required  by  the  rule  for  multiplication,  and  sepa- 

-r rate  the  dollars  from  the  cents  in  the  product  by  a 

$985.50  decimai  pomt. 


Multiply— 

2.  $61.34  by  8. 

3.  $40.65  by  11. 

4.  $3,125  by  25. 

5.  $19.06  by  12. 

6.  $20,013  by  13. 


7.  $93.56  by  100. 

8.  $1005  by  13. 

9.  $1056  by  171. 

10.  $34,055  by  1000. 

11.  $103.03  by  100. 


12.  $13.06  X  10  X  3  =  what  amount? 

13.  What  will  it  cost  to  build  7  cottages,  at  $2500.50 
each  ? 

14.  If  a  man  earns  $125.87  each  month,  how  much 
does  he  earn  in  12  months? 

15.  How  much  will  640  acres  of  land  cost,  at  $120 
per  acre  ? 


124 


UNITED   STATES  MOSEY. 


DIVISION. 


258. — 1.  How  many  writing-books,  at  12  cents  each, 
can  be  bought  for  $1.44  ? 

2.  At  25  cents  each,  how  many  spelling-books  can  be 
bought  for  $1.25? 

3.  James  paid  $3.75  for  3  hats,  how  much  were  they 
apiece  ? 

4.  At  3  dimes  a  yard,  how  many  yards  of  cloth  can 
be  bought  for  $2.10  ? 

WRITTEN  EXERCISES. 

259.— 1.  What  is  the  quotient   of  $985.50  divided 

by  9? 

Solution. — Write  the  numbers  and  divide, 
9 J$ 985.50       as  required  by  the  rule  in  division,  and  separate 
$109.50        the  dollars  from  the  cents  in  the  quotient  by  a 
decimal  point. 

2.  What  is  the  cost  of  a  barrel  of  flour  when  14  bar- 
rels cost  #Qi  9 


Solution. — Continue  the  division 
after  dividing  the  dollars,  supplying 
the  orders  of  cents  in  the  dividend  by 
ciphers. 


W$91.00($6.50 

84 
70 

70 


0 
Divide — 

3.  $325.20  by  8. 

4.  $67.10  by  11. 

5.  $626.50  by  25. 

6.  $73.44  by  12. 

7.  $9356  by  100. 

8.  $13065  by  13. 


.56  by  4. 

10.  $634055  by  1000. 

11.  $1135  by  20. 

12.  $19.65  by  15. 

13.  $114.24  by  6. 

14.  $111144  by  100. 


UNITED  STATES  MONEY.  125 

15.  If  11  tons  of  hay  cost  $184.25,  what  is  the  cost 
of  1  ton  ? 

16.  At  $6.50  a  barrel,  how  many  barrels  of  flour  can 
be  bought  for  $91  ? 

650)9 100(  14  Solution.— Prepare  the  numbers  for  di- 

^'9_r_  viding  by  reducing  both  to  cents.    $6.50  — 

2600  650  cents;   and  $91 -=9100  cents;   9100 -h- 

2600  ^50  =  14,  the  result  required. 

17.  How  many  tons  of  coal,  at  $5.25  a  ton,  can  be 
bought  for  $105  ? 

18.  When  20  tons  of  coal  can  be  bought  for  $105, 
what  is  the  cost  of  1  ton  ? 

19.  At  80  cents  a  bushel,  how  many  bushels  of  corn 
can  be  bought  for  $68  ? 

20.  Jones  paid  for  his  farm  $10564,  and  Smith  bought 
some  land  at  one-fourth  of  that  sum.  What  was  the 
cost  of  Smith's  land  ? 

21.  One  half  of  Jones's  farm  is  320  acres;  if  the 
whole  cost  him  $7680,  what  was  the  price  per  acre  ? 

ALIQUOT    PARTS. 

260.— 1.  What  part  of  a  dollar  is  50  cents? 

2.  What  part  of  a  dollar  is  25  cents  ? 

3.  How  many  cents  in  half  a  dollar?  In  a  fourth  of 
a  dollar  ? 

4.  What  part  of  a  dollar  is  obtained  by  dividing  it 
by  2  ?     By  dividing  it  by  4  ? 

5.  What  part  of  a  dollar  is  20  cents?     Is  10  cents? 

6.  How  many  cents  in  a  fifth  of  a  dollar?  In  a  tenth 
of  a  dollar? 

n  * 


126  UNITED   STATES  MONEY. 

7.  What  part  of  a  dollar  is  obtained  by  dividing  it  by 
5  ?     By  dividing  it  by  10  ? 

8.  What  part  of  a  dollar  is  33^  cents?  16-|  cents? 
8|  cents  ? 

9.  How  many  cents  in  one  third  of  a  dollar  ?  In  one 
sixth  of  a  dollar  ?  In  one  seventh  of  a  dollar  ?  In  one 
twelfth  of  a  dollar  ? 

10o  What  part  of  a  dollar  is  obtained  by  dividing  it 
by  3  ?     By  dividing  it  by  6  ?     By  dividing  it  by  8  ? 

11.  What  part  of  a  dollar  is  12£  cents?  Is  37^  cents? 
Is  621  Cents  ?     Is  87£  cents  ? 

12.  How  many  cents  in  one  eighth  of  a  dollar?  In 
five  eighths  ?     In  seven  eighths  ? 

13.  What  part  of  a  dollar  is  66f  cents?  Is  75  cents  f 
Is  83^  cents  ? 

14.  What  will  25  yards  of  cloth  cost  at  12 \  cents  a 
yard? 

Solution.— If  1  yard  cost  12J  cents,  or  \  of  a  dollar,  2o 
yards  will  cost  25  times  \  of  a  dollar,  or  2f  of  a  dollar,  which  are 
$3J,  or  $3.12i. 

15.  What  will  60  pairs  of  hose  cost  at  25  cents  a  pair? 

16.  What  will  37  bushels  of  apples  cost  at  33^  cents 
a  bushel  ? 

17.  How  much  must  be  paid  for  30  bushels  of  corn 
at  66%  cents  a  bushel  ? 

Solution.— If  66f  cents,  or  f  of  a  dollar,  must  be  paid  for 
1  bushel,  there  must  be  paid  for  30  bushels,  30  times  f  of  a 
dollar,  or  %°  of  a  dollar,  which  are  $20. 

18.  How  much  must  be  paid  for  16  baskets  of  peaches 
at  87^  cents  a  basket? 

19.  At  75  cents  each,  what  will  48  arithmetics  cost? 


UNITED    STATES    M02^FT 


127 


DEFINITION. 

261.  An  Aliquot  Part  of  a  number  is  any  exact  frac- 
tional part  of  that  number. 

In  business,  frequent  use  is  made  of  the  convenient 
aliquot  parts  of  a  dollar,  given  in  the  following — 


TABLE. 
10  cents  are  —  of  $1. 

20    "         "     j  of  $1. 

25    «         «     7  of  01. 

4 

50    «         >>     I  of$l. 


8-  cents  are  —  of  . 

o  12 


mi 


3 

33± 


jof$l. 

lof$l. 

Uf$i. 


WRITTEN  EXERCISES. 

262. — 1.  What  is  the  cost  of  49  yards  of  cloth  at 

$1.75  a  yard? 

Solution. 

At  $1.00  per  yard,  the  cost  of  49  yards  is  $49. 

a       .50     <<        «         «        "        "  ~2  of  $49 ==  24-50 

"        .25     "        "  "        "        "  7  °f  $4-9  ^^  12.25 

"  $1.75     «        «  "        "        "  $85.75 

2.  How  much  must  be  paid  for  54  bushels  of  wheat 
at  $1.66|  per  bushel? 

3.  What  will  72  hats  cost  at  87£  cents  each? 

4.  What  will   96  pounds  of  rice  cost  at  8£  cents  a 
pound  ? 

263.  Rule  for  Finding  the  Cost  of  Articles  by  Aliquot  Parts.- 
First  find  the  cost  at  $1,  and  then  take  the  aliquot 
parts  of  this  amount. 


128  UNITED    STATES    MONEY. 

PROBLEMS. 

264. — 1.  What  is  the  value  of  758  yards  of  carpeting 
at  $1.25  per  yard? 

2.  How  much  must  be  paid  for  178  yards  of  gingham 
at  37^  cents  per  yard  ? 

3.  What  will  1840  tons  of  coal  cost  at  $4.62*  per  ton? 

4.  At  33^  cents  a  pair,  what  is  the  cost  of  96  pairs  of 
gloves  ? 

BILLS. 

265.  A  Bill  of  Goods  is  a  written  statement  of  articles 
sold,  the  quantity  and  price  of  each  article,  and  the  en- 
tire cost  of  the  whole. 

266.  A  Bill  of  Services  is  a  written  statement  of  labor 
performed,  the  time,  kind  and  value  of  such  services. 

267.  A  Debtor  is  the  party  who  owes  a  bill,  and  a 
Creditor  is  the  party  to  whom  the  bill  is  owed. 

268.  A  bill  is  Receipted  when  the  creditor,  or  some 
one  acting  for  him,  acknowledges  its  payment  in  writing. 


269.  Test  Questions.— 1.  How  do  you  write  United  States 
money  for  adding  ?     For  subtracting  ? 

2.  How  do  you  write  the  numbers  and  multiply  in  United 
States  money?     How  do  you  write  the  numbers  and  divide? 

3.  In  dividing  dollais,  when  there  is  a  remainder,  how  may 
you  continue  the  division?  When  the  divisor  expresses  cents 
and  the  dividend  dollars,  how  do  you  prepare  the  numbers  for 
dividing? 

4.  What  are  aliquot  parts  of  a  number?  How  do  you  com- 
pute the  cost  of  articles  by  aliquot  parts? 

5.  What  is  a  bill  of  goods?  A  bill  of  services?  When  is  a 
bill  receipted? 


UNITED  STATES  MONET. 


129 


WRITTEN   EXERCISES. 

270.  Copy    the    following    bills,    and    compute   the 
amount  due — 

Bill  Unreceipted. 

Philadelphia,  Jan.  4,  1871. 
John  W.  Brewster, 

Bought  of  William  Collins  &  Co. 


18  yards  of  Muslin,  @ 

$.16l 

$  3 

00 

28  yards  of  Cambric,  @ 

.25 

7 

00 

12  dozen  Napkins,      @ 

2.12\ 

25 

50 

6  dozen  Towels,        @ 

$5.25 

31 

50 

$67 

00 

The  character  @  signifies  at.    Thus,  18  yards  of  muslin  @ 
$.16|  means  18  yards  of  muslin  at  $.16f  per  yard. 


Bill  Receipted. 

Portland,  April  3,  1871. 
Henry  C.  Warren, 

Bought  of  Hamstead  Brothers. 


10  pounds  Oolong  Tea,  @  $1.20 
20  pounds  Rio  Coffee,      @       .37  ~ 
15  pounds  Sugar,  @       .  12- 


Received  payment, 

Hamstead  Brothers. 


130 


UNITED  STATES  MONEY. 


Bill  Receipted  by  Clerk. 

Hartford.  March  11,  1871. 
Col.  Ambrose  Chase, 

To  William  T.  Stone,  Dr. 


1871. 

Feb. 

3 

For  Labor  on  wall,  9  days,  @  $3.50 

$31 

50 

tt 
March 

10 

5 

"   Excavating  eellar, 

"   Labor,    laying    stone,     8    days, 

15 

50 

@  $3.25, 

26 

00 

$73 

00 

Received  payment, 

William  T.  Stone, 

per  M.  T.  Sneider. 


Bill  tvith  Credit  Items. 

Providence,  April  27,  1871. 
Mr.  Walter  Bowen, 

To  John  Burgess  &  Sons,  Dr. 


1871. 

Jan. 

April 

17 
8 

25 

17 

24 

To  72  tons  Coal,           @  $8.62 j 
"  15  cords  Oak  Wood,  @   7.25 
"  18  cords  Pine  Wood,®  6.33J 

Or. 

By  Merchandise,  as  by  his 

bill,                             $550.50 
By  Cash,                          200.00 

$621 
108 
114 

00 

75 
00 

Feb. 

U 

$843 
750 

75 
50 

Balance  due  J.  B.  &  Sons 

$  93 

25 

DENOMINATE  NUMBERS. 


131 


SECTION    XXIII. 
MEASURES    OF  EXTENSION. 

LINEAR   MEASURES. 


271.  Linear  or  Long 
Measures  are  those  used 
in  ascertaining  dis- 
tances and  the  dimen- 
sions of  things. 

The  units  of  length 
are  an  inch,  a  yard,  a 
rod  and  a  mile. 

One  Inch. 


TABLE. 

12  inches  (in J  are  1  foot ft. 

3  feet  "    1  yard yd,. 

5%  yards  "    1  rod rd. 

320  rods  "    1  mile.  . . .  mi. 

1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63360  in. 

Also,  4  inches  are  1  hand,  used  in  measuring  the  height  of 
horses;  3  feet  are  1  pace;  and  6  feet  are  1  fathom,  used  in 
measuring  depth  of  water. 

272.  In  Cloth  Measure  the  yard  is  divided  into  halves, 
quarters,  eighths  and  sixteenths. 

273.  The  Surveyor's  Chain,  called  Gunters  chain,  used 


132  DENOMINATE  NUMBERS. 

in  measuring  roads  and  boundaries  of  land,  is  4  rods  in 
length,  and  is  divided  into  100  links. 

Thus,  7T9o%  inches  are  1  link ;  100  links,  or  4  rods,  are  1  chain; 
and  80  chains  are  1  mile. 

274. — 1.  How  many  inches  are  2  feet?     Are  6  feet? 

2.  How  many  inches  is  1  yard  ?     Are  2  yards  ? 

3.  How  many  feet  are  6  yards  ?     Are  8  yards  ? 

4.  How  many  feet  in  24  inches?     In  72  inches? 

5.  How  many  yards  in  36  inches  ?      How  many  feet 
in  72  inches  ?     How  many  yards  in  72  inches  ? 

6.  How  many  yards  in  24  feet?     In  36  feet? 

7.  How  many  yards  in  3  rods  ?     In  5  rods  ? 

8.  In  1  mile  how  many  rods  ?     In  \  of  a  mile  how 
many  rods  ? 

Solution. — Since  in  1  mile  there  are  320  rods,  in  \  of  a  mile 
there  must  be  one  half  of  320  rods,  or  160  rods. 

9.  How  many  rods  in  \  of  a  mile?    In  i  of  a  mile? 

10.  How  many  inches  in  3  quarters  of  a  yard  ? 

11.  How  many  eighths  in  3  quarters  of  a  yard? 

12.  How  many  yards  in  33  feet?     In  45  feet? 

13.  How  many  rods  in  33  feet?     In  66  feet? 

14.  What  part  of  a  foot  is  4  inches?    Is  8  inches? 

15.  At  5  cents  a  foot,  what  will  10^  yards  of  wire 
cost? 

16.  When  ribbon  is  worth  20  cents  a  yard,  what  are 
3  yards  and  3  quarters  worth  ? 

17.  What    part   of    a   yard    is    9    inches?      Is    45 
inches  ? 

18.  At  \  of  a  dollar  a  rod,  how  much  will  it  cost  to 
construct  a  path  \  of  a  mile  long  ? 


DENOMINATE  NUMBERS.  133 

DEFINITIONS. 

275.  Denomination  is  the  name  of  the  unit  expressing  a 
measure  or  number. 

Of  two  denominations,  the  higher  is  that  which  expresses  the 
greater  value,  and  the  lower  is  that  which  expresses  the  less 
value. 

276.  A  Denominate  Number  is  a  number  expressed  in 
one  or  more  denominations. 

WRITTEN  EXERCISES. 

277. — 1.  How  many  feet  are  46  rods  ? 

Solution. — Since  1  rod  is  5J  yards,  there  must 
be  5J  times  as  many  yards  as  rods ;  hence,  in  46  rods 
there  must  be  5J  times  46  yards,  or  253  yards. 

Since  1  yard  is  3  feet,  there  must  be  3  times  as 
many  feet  as  yards ;  hence,  in  253  yards,  there  are 
3  times  253  feet,  or  759  feet,  which  is  the  answer  re- 
quired. 
759  ft 

2.  How  many  rods  are  759  feet  ? 

3  ft  )  759  ft  Solution. — Since  3  feet  are  one 

2  yard,  759  feet  must  be  as  many  yards 

&J  yd.  )  253  yd.  as  there  are  times  3  feet  in  759  feetj 

2_  _2_  which  are  253  times.     Hence,  759 

1 1  hf.  yd.)  506  hf.  yd.        feet  =  253  yards- 

~~7r    j  Since  5J  yards  are  1  rod,  253  yards 

4-     r  >  must  be  as  many  rods  as  there  are 

times  5 \  yards  in  253  yards,  or  times  11  half  yards  in  506  half 

yards,  which  are  46  times.    Hence,  253  yards,  or  759  feet  =  46  rods6 

3.  How  many  feet  are  80  rods?     Are  15  miles? 

4.  How  many  rods  are  1320  feet?     Are  4700  feet? 

5.  How  many  miles  are  79200  feet  ?   Are  7920  chains  ? 

12 


46  rd. 

230 

23 

253  yd. 
3 

134  DENOMINATE    NUMBERS. 

SURFACE   MEASURES. 

278.  A  Surface  is  that  which  has  length,  and  breadth 
or  width,  without  thickness. 

279.  A  Square  is  a  figure  having  four  equal  straight 
sides,  and  four  equal  corners  or  angles. 

280.  A  Square  Iuch  is  a  square 
having  each  of  its  sides  1  inch  in 
length. 

281.  A  Square  Foot  is  a  square 
having  each  of  its  sides  1  foot  in 
length. 


282.  A  Square   Rod   is   a   square  i  square  inch. 
having  each  of  its  sides  1  rod  in  length. 

283.  A  Square  Mile  is  a  square  having  each  of  its  sides 
1  mile  in  length. 

281.  Surface  Measures  are  those  used  in  ascertaining 
extent  of  surface. 

The  units  of  surface  are  a  square  inch,  a  square  foot,  a 
square  yard,  a  square  rod,  an  acre  and  a  square  mile. 

TABLE. 

144  square  inches  (sq.  in.)  are  1  square  foot .  ..sq.  ft. 
9  square  feet  "    1  square  yard .  sq.  yd, 

30^  square  yards  "    1  sq.  rod  or  perch.  .  P, 

160  sq.  rods  or  perches  "  1  acre ,  A 

GJfO  acres  "  1  square  mile  . .  sq.  n%a 

1  A  =  160  P.  =  4840  sq.  yd.  =  43560  sq.  ft.= 
6272640  sq.  in. 

285.  In  the  Measurement  of  Land,  16  square  rods  are 
1  square  chain  (sq.  ch.),  and  10  square  chains  are  1  acre. 


DENOMINATE  NUMBERS. 


135 


286. — 1.  How  many  square  inches  in  2  square  feet? 

2.  How  many  square  feet  in  4  square  yards  ?     In  8 
square  yards  ?     In  2  square  rods  ?     In  3  square  rods  ? 

3.  How  many  square  yards  in  36  square  feet  ?   In  72 
square  feet  ? 

4.  How  many  square  rods  in  1  acre  ?    In  |-  of  an  acre  ? 

5.  What  part  of  an  acre  is  80  square  rods  ?     Is  40 
square  rods? 

6.  How  many  square  rods  are  27  square  feet? 

7.  How  many  square  rods  are  60|  square  yards  ? 

8.  What  part  of  an  acre  is  5  square  chains  ? 

9.  What  will  it  cost  to  pave  108  square  feet  of  a  walk, 
at  50  cents  a  square  yard  ? 


WMITTEN  EXERCISES. 


287.— 1.  How  many 
6quare  feet  in  25  acres  ? 

Solution. 

25  A. 

160 
1500 
25 
4000  P. 

soj 

120000 

1000 
121000  sq.  yd. 

9 


1089000  sq.ft. 


2.     How     many    acres     in 
1089000  square  feet  ? 

Solution. 
9  sq.ft.)  1089000  sq.ft. 
30J  sq.  yd.   J 121000  sq.  yd. 


121  fourths  ) 484000  fourths 
sq.  yd.        sq.  yd. 

160  P.) 4000  P. 

25  A. 

3.  How  many  square  inches 
in  15  square  rods  ? 

4.  How  many  square   rods 
in  588060  square  inches  ? 


136 


DENOMINATE    NUMBERS. 


5.  In  100  square  chains  how  many  square  rods  ? 

6.  In  1600  square  rods  how  many  square  chains  ? 

7.  In  36  square  miles  how  many  square  rods  ? 

8.  In  23040  square  rods  how  many  square  miles? 

9.  In  120  square  yards  how  many  square  inches? 

10.  How  many  acres  are  there  in  a  lot  of  land  80  rods 
long  and  72  rods  wide  ? 

11.  How  many  acres  of  land  are  there  in  a  road  6^ 
miles  long  and  5  rods  wide  ? 


CUBIC   MEASURES. 

288.  A  Solid,  or  Volume,   is  that  which    has  length, 
breadth  and  thickness,  or  depth. 

289.  A  Cube  is  a  solid  bounded  by  six  equal  squares, 
called  faces. 

290.  A  Cubic  Inch  is 
a  cube  whose  faces  are 
each  1  inch  square. 

291.  A  Cubic  Foot  is 
a  cube  whose  faces  are 
each  1  foot  square. 

292.  A  Cubic  Yard  is 
a  cube  whose  faces  are 
each  1  yard  square. 

293.  Cubic  Measures 
are  those  used  in  meas- 
uring things  that  have 
length,  breadth  and 
depth,  or  thickness. 

The  unite  are  a  cubic  inch,  a  cubic  foot  and  a  cubic 
yard;  also,  a  cord  foot  and  a  cord. 


A  Cubic  Inch. 


DENOMINATE  NUMBERS. 


137 


TABLE. 

1728  cubic  inches  (cu.  in.)  are  1  cubic  foot. . .  cu.  ft. 
27  cubic  feet  "    1  cubic  yard. .  cu.  yd. 

Also, 

16  cubic  feet  are  1  cord  foot cd.  ft. 

8  cord  feetf  or  )  ^  1  CQrd cd 

128  cubic  feet,      ) 

1  cu.  yd.  -  27  cu.  ft.  =  46656  cu.  in. 


294.  Wood  as  usually  cut  for  the  market  is  4  feet  long, 
and  is  piled  in  ranges  4  feet  high.  Of  such  ranges,  a 
part  that  is  1  foot  of  the  length  of  the  range  is  1  cord 
foot,  or  1  foot  of  wood;  and  a  part  that  is  8  feet  of  the 
length  of  the  range  is  1  cord  of  wood. 

295. — 1 .  How  many  cubic  inches  in  2  cubic  feet  ? 

2.  How  many  cubic  feet  in  2  cubic  yards  ? 

3.  How  many  cubic  feet  in  2  cord  feet  ?  In  3  cord 
feet  ?    In  4  cord  feet  ?    In  |  of  a  cord  ?    In  |  of  a  cord  ? 

4.  How  many  cords  in  48  cord  feet?   In  88  cord  feet? 


12* 


138 


DENOMINATE  NUMBERS. 


WRITTEN   EXERCISES. 


296. — 1.  How  many  cubic 
inches  in  306  cubic  yards  ? 
Solution. 


2.  How  many  cubic  yards 
in  14276736  cubic  inches? 


306  cu.  yd. 

27 


2142 
612 

8262  cu.  ft. 
1728 


66096 
16524 
57834 

8262 

14276736  cu.  in. 


Solution. 
1728)14276736(8262  cu.fi. 
13824 

4527 
3456 

10713 
10368 


3456 
3456 


27)8262(306  cu.  yd. 
81 
162 
162 


3.  How  many  cubic  feet  in  365  cords  ? 

4.  How  many  cords  in  46720  cubic  feet? 


297.  Test  Questions. — 1.  For  what  is  linear  or  long 
measure  used  ?     What  are  its  units  ?     Recite  the  table. 

2.  How  is  the  yard  divided  in  cloth  measure?  How  many 
inches  in  1  hand?  How  many  feet  in  1  pace?  In  1  fathom? 
What  is  the  length  of  Gunter's  chain  ? 

3.  What  is  a  denomination  ?  Of  two  denominations,  which 
is  the  higher?    The  lower?     What  is  a  denominate  number? 

4.  What  is  a  surface  ?  A  square  ?  A  square  inch  ?  A  square 
foot?     A  square  rod?    A  square  mile?     Recite  the  table. 

5.  What  is  a  volume  or  solid?  A  cube?  A  cubic  inch?  A 
cubic  foot  ?  A  cubic  yard  ?  For  what  are  cubic  measures  used  ? 
What  are  the  units  of  cubic  measures  ?  Recite  the  table.  How 
is  wood  usually  cut  and  ranged  ? 


DENOMINATE  NUMBERS.  139 

SECTION    XXIV. 
MEASURES  OF  CAPACITY. 

LIQUID    MEASURES. 

298.  Liquid  Measures  are  used  in  measuring  liquids. 
The  units  are  a  gill,  a  pint,  a  quart  and  a  gallon. 

TABLE. 

J^  gills  (gi.)  are  1  pint pt. 

2  pints  "    1  quart.  .  .  .  qt. 

Jj,  quarts         "    1  gallon  .  .  .  gal. 

1  gal.  =4qi.  =  8  pt.  =  32  gi. 

A  Barrel,  regarded  as  a  measure  of  cisterns,  vats,  etc., 
is  31  \  gal.,  and  a  Hogshead  is  63  gallons. 

A  Gallon,  liquid  measure,  contains  231  cubic  inches. 

299. — 1.  How  many  gills  in  8  pints?     In  12  pints? 

2.  How  many  pints  in  32  gills?     In  48  gills? 

3.  How  many  pints  in  8  quarts  ?     In  1 1  quarts  ? 

4.  How  many  quarts  in  1 6  pints  ?     In  24  pints  ? 

5.  How  many  quarts  in  6  gallons?     In  10  gallons? 

6.  How  many  gallons  in  24  quarts  ?     In  48  quarts  ? 

7.  What  part  of  a  quart  is  2  gills  ?     Is  6  gills  ? 

8.  What  part  of  a  gallon  is  1  quart?     Is  3  quarts? 

9.  At  40  cents  a  gallon,  what  will  3  quarts  of  wine  cost  ? 

10.  How  many  pint-and-a-half  bottles  can  be  filled 
from  a  gallon  and  a  half  ? 

11.  How  many  barrels  can  be  filled  from  fifty  hogs- 
heads ? 

12.  How  many  quarts  are  there  in  a  hogshead  of 
molasses? 


140 


DEXOMIXA  TE  NUMBERS. 


WRITTEN   EXERCISES. 


300. — 1.  How  many  gills 
in  84  gallons  ? 

Solution. 

84  gal. 

4 

336  qt. 

2 
672  pt 

4 


2.  How  many  gallons  in 
2688  gills  ? 

Solution. 

4)2688  gi. 
2)672  pt 
4)336  qt 
84  gal. 


2688  gi. 

3.  How  many  pints  in  116  hogsheads? 

4.  How  many  hogsheads  in  928  pints  ? 

5.  At  3  cents  a  gill,  what  will  a  barrel  of  liquor  cost  ? 

6.  At  6  cents  a  quart,  how  much  must  be  paid  for  40 
gallons  of  milk  ? 


DRY   MEASURES. 

301.  Dry  Measures  are  those  used  in  measuring  grain, 
fruit,  coal,  salt,  and  similar  articles. 

The  units  are  a  pint,  a  quart,  a  peck  and  a  bushel. 

TABLE. 

2  pints  (pt.)  are  1  quart.  .  .  qt. 
8  quarts  "  1  peck ....  ph. 
4  peeks  "    1  bushel .  .  bu. 

lbu  =  4pk.=  32  qt.  =  64  pt. 

A  Gallon,  or  4  quarts,  dry  measure,  contains  268| 
cubic  inches;  and  a  bushel,  2150^^  cubic  inches. 


DENOMINATE  NUMBERS.  Ill 

302. — 1.    How  many   pints    in    6    quarts  ?      In    12 
quarts  ? 

2.  How  many  quarts  in  8  pecks?     In  11  pecks  ? 

3.  How  many  quarts  in  12  pints?     In  24  pints  ? 

4.  How  many  pecks  in  64  quarts?    In  88  quarts? 

5.  How  many  pecks  in  9  bushels?     In  12  bushels? 

6.  How  many  bushels  in  36  pecks?     In  60  pecks  ? 

7.  What  will  2\  quarts  of  chestnuts  cost,  at  5  cents  a 
pint  ? 

8.  What  will  3f  gallons  of  milk  cost,  at  4  cents  a 
quart  ? 

9.  What  will  a  peck  and  a  half  of  berries  cost,  at 
16f  cents,  or  ^  of  a  dollar,  a  quart  ? 

WRITTEN  EXERCISES. 

303. — 1.  In  310  bushels,  how  many  quarts? 

2.  In  9920  quarts,  how  many  bushels? 

3.  How  many  pints  in  64  pecks  ? 

4.  How  many  bushels  in  40960  pints  ? 

5.  At  8  cents  a  quart,  how  much  are  6  bushels  of 
chestnuts  worth? 

6.  At  8  cents  a  quart,  how  many  bushels  of  chestnuts 
can  be  bought  for  $15.36  ? 


304.  Test  Questions.  —  1.  What  are  liquid  measures? 
Name  the  units  of  liquid  measures.     Kecite  the  table. 

2.  How  many  gallons  in  a  barrel  when  it  is  regarded  as  a 
measure?  A  hogshead?  How  many  cubic  inches  does  a  gallon 
contain  ? 

3.  What  are  dry  measures?  Name  the  units  of  dry  measures. 
Recite  the  table. 

4.  How  many  cubic  inches  does  a  gallon  contain?  How  many 
does  a  bushel  contain  ? 


142 


DENOMINATE  NUMBERS. 


SECTION  XXV. 
MEASURES  OF  WEIGHT. 


WOIRDUPOIS   WEIGHTS. 


305.  Avoirdupois  Weights  are  those  used  in  weighing 
produce,  groceries,  coal,  iron,  and  similar  articles. 

The  units  are  an  ounce,  a  pound,  a  hundred-weight  and 
a  ton. 

TABLE. 

16  ounces  (oz.)  are  1  pound lb. 

100  pounds  "     1  hundred-weight . .  cwt. 

20  hundred-weight   "     1  ton T. 

IT  =20  cwt.  =  2000  lb.  =  32000  oz. 
Also,  56  pounds  of  corn  or  rye,  60  pounds  of  wheat  or  pota- 
toes, or  32  pounds  of  oats,  are  1  bushel;    100  pounds  of  dry- 
fish  are  1  quintal ;  100  pounds  of  grain  are  1  cental;  and  196 
pounds  of  flour,  or  200  pounds  of  beef  or  pork,  are  1  barrel. 

306.  At  the  Custom  Houses,  in  collecting  duties  on 
English  goods,  and  in  the  wholesale  and  freighting  of 
coal,  28  pounds  are  1  quarter,  4  quarters,  or  112  pounds, 
are  1  hundred-weight,  and  20  hundred-weight,  or  2240 
pounds,  are  1  ton,  called  the  long  ton. 


DENOMINATE  NUMBERS.  143 

307. — 1.    How  many   ounces  in    2    pounds?      In    3 
pounds  ?     In  5  pounds  ? 

2.  How  many  pounds  in  32  ounces  ?     In  48  ounces  ? 

3.  How  many  pounds  in  6  hundred- weight?     In  9 
hundred-weight  ? 

4.  How  many  hundred- weight  in  500  pound??     In 
700  pounds  ? 

5.  How  many  tons  in  50  hundred- weight  ?     In  100 
hundred-weight  ? 

6.  What  part  of  a  pound  is  12  ounces? 

7.  What  part  of  a  hundred- weight  is  25  pounds  ?    Is 
75  pounds  ? 

8.  How  many   pounds  in   1   quarter  of  a  hundred- 
weight ?     In  3  quarters  of  a  hundred-weight  ? 

9.  At  $2  a  hundred-weight,  what  will  1^  tons  of  iron 
cost? 

10.  At  10  cents  an  ounce,  what  will  1^  pounds  of 
rhubarb  cost  ? 

11.  At  $1.50  per  bushel,  how  much  must  be  paid  for 
a  bag  of  wheat  weighing  90  pounds  ? 

WRITTEN  EXERCISES. 

308. — 1.  How  many  ounces  in  316  pounds? 

2.  How  many  pounds  in  65  tons  ? 

3.  How  many  pounds  in  5056  ounces  ?     How  many 
tons  in  13000  pounds  ? 

4.  How  many  bushels  in  a  load  of  wheat  weighing 
4590  pounds? 

5.  What  is  the  weight  of  76|  bushels  of  wheat  ? 

6.  At  H  cents  per  pound,  what  will  be  the  cost  of 
freighting  16  tons  of  goods? 


144 


DENOMINATE  NUMBERS. 
TROY    WEIGHTS. 


309.  Troy  Weights 

are  those  used  in 
weighing  gold,  sil- 
ver and  gems. 

The  units  are  a 
grain,  a  pennyweight, 
an  ounce  and  a 
pound. 


TABLE. 

2Jf  grains  (gr.)    are  1  pennyweight ....  pwt. 

20  pennyweights  "     1  ounce oz. 

12  ounces  "    1  pound lb. 

1  lb.  =  12  oz.  =  240  pwt.  =  5760 gr. 
A  Pound  Avoirdupois  is  equal  to  7000  Troy  grains. 

310.  Apothecaries,  in  compounding  medicines  and  in 
putting  up  prescriptions,  either  use  only  the  denomina- 
tions of  grains,  ounces  and  pounds,  or  subdivide  the 
Troy  pound  into  grains,  scruples,  drams  and  ounces* 
Thus, 

20 grains  (gr.)  are    1  scruple  ....  9. 

S  scruples  "     1  dram 3. 

8  drams  "     1  ounce §. 

12  ounces  "     1  pound lb. 

311. — 1.  How  many  grains  in  1  pennyweight?    In  2 

pennyweights  ? 

2.  How  many  pennyweights  in  48  grains  ? 


DENOMINATE  NUMBERS.  145 

3.  How  many  pennyweights  in  2  ounces?  In  5 
ounces  ? 

4.  How  many  ounces  in  40  pennyweights  ?  In  100 
pennyweights? 

5.  How  many  ounces  in  3  pounds  ?     In  10  pounds? 

6.  How  many  pounds  in  48  ounces  ?    In  144  ounces  ? 

7.  At  $3.25  a  pennyweight,  what  is  the  value  of  a 
jewel  weighing  12  pennyweights? 

8.  At  3  cents  a  scruple,  what  is  the  value  of  a  drug 
weighing  1  ounce? 

9.  How  many  doses,  of  5  grains  each,  are  there  in 
1  dram  of  medicine  ? 

WRITTEN  EXERCISES. 

312. — 1.  How  many  grains  in  65  ounces  ? 

2.  How  many  ounces  in  31200  grains? 

3.  How  many  pennyweights  in  51^  pounds  ? 

4.  How  many  pounds  in  12360  pennyweights? 

5.  At  25  cents  a  dram,  what  will  80  pounds  of  drugs 
cost? 

6.  At  25  cents  a  dram,  how  many  pounds  of  drugs 
may  be  bought  for  $1920  ? 

7.  If  the  gold  coin  of  the  United  States  is  composed 
of  9  parts  of  pure  gold  and  1  part  of  alloy,  how  many 
pennyweights  of  alloy  are  there  in  20  pounds  of  coin  ? 


313.  Test  Questions.— 1.   What  are  avoirdupois  weights? 
Name  the  units  of  avoirdupois  weights.     Recite  the  table. 

2.  How  many  pounds  of  corn  or  rye  are  1  bushel  ?      How 
many  pounds  of  wheat  or  potatoes  ?     How  many  of  oats? 

3.  How  many  pounds  of  fish  are  1  quintal?      How  many 

13 


146  DENOMINATE    NUMBERS. 

pounds  of  grain  are  1  cental  ?     How  many  pounds  of  flour  are 
1  barrel  ?     How  many  pounds  of  beef  or  pork  are  1  barrel  ? 

4.  How  many  pounds  are  a  quarter  in  the  wholesale  and 
freighting  of  coal?  How  many  pounds  are  a  hundred-weight? 
How  many  pounds  in  a  ton  ? 

5.  What  is  Troy  weight?  What  are  the  units  of  Troy  weight? 
Recite  the  table.     Recite  the  table  of  apothecaries'  weights. 

6.  How  many  Troy  grains  are  1  pound  Troy  ?  How  many 
Troy  grains  are  equal  to  1  pound  avoirdupois  ?  Which  is  the 
heavier,  a  pound  Troy  or  a  pound  avoirdupois  ? 


SECTION   XXVI. 

CIRCULAR  MEASURES. 

314.  A  Circle  is  a  plane  surface  bounded  by  a  line,  all 
parts  of  which  are  equally  distant 
from    a    point    within   called    the 
center. 

315.  A  Circumference  is  the  line 
that  bounds  a  circle. 

316.  An  Arc  is  any  part  of  the 
circumference ;  as  AD,  or  DB. 

317.  A  Degree  is  one  of  the  360  equal   parts  of  a 
circumference. 

318.  An  Angle  is  the  differ- 
ence of  direction  of  two  lines 
which  meet  at  a  point. 

Thus,  the  lines  AB  and  A  C,  which     A^-  —c 

meet  at  A,  form  the  angle  BAC. 

319.  The  Measure  of  an  angle  whose  sides  meet  at 


DENOMINATE  NUMBERS. 


147 


the  center  of  a  circle  is  that  part  of  the  circumference 
between  the  sides. 

Thus,  the  arc  AD  is  the 
measure  of  the  angle  A  CD 
which  is  formed  in  the  circle, 
(page  146). 

320.  Circular  Measures 

are  those  used  in  meas- 
uring the  arcs  of  circles, 
angles  and  the  difference 
of  directions. 

The  units  are  a  second, 
a  minute,  a  degree  and  a 
circumference. 

TABLE. 

60  seconds  (")  are  1  minute '. 

60  minutes  "    1  degree °. 

360  degrees  "    1  circumference . .  C. 

1C.  =  360°  =  21600'  =  1296000". 

321.  A  Minute  of  the  circumference  of  the  earth,  or  a 
geographic  mile,  is  about  \\  common  miles. 

322. — 1.  How  many  seconds  in   2   minutes?     In    3 
minutes  ?     In  5  minutes  ? 

2.  How   many   minutes   in   120  seconds?      In    180 
seconds  ?     In  300  seconds  ? 

3.  How  many  minutes  in  3  degrees?     In  4  degrees? 

4.  How  many  degrees    in    120    minutes?      In    240 
minutes  ?     In  360  minutes  ? 

5.  What  part  of  a  circumference  is  30  degrees  ? 

6.  How  many  degrees  in  \  of  a  circumference  ? 


148  DENOMINATE  NUMBERS. 

WRITTEN  EXERCISES. 

323. — 1.  How  many  seconds  in  240  degrees? 

2.  How  many  degrees  in  864000  seconds  ? 

3.  How  many  minutes  in  f  of  a  circumference? 

4.  How  many  degrees  in  14200  minutes? 

5.  How  many  seconds  in  §  of  an  hour  ? 

6.  If  a  vessel  sail  2|  degrees  of  the  circumference 
of  the  earth  in  one  day,  in  how  many  days  will  it  sail 
180  degrees  ? 

7.  In  sailing  180  degrees  of  the  circumference  of  the 
earth,  how  many  common  miles  does  a  vessel  pass  over? 


SECTION    XXVII. 

MEASURES  OF  TIME. 

324.  Measures  of  Time  are    those    used  in    measuring 
time  or  duration. 

The  units  are  a  second,  a  minute,  an  hour,  a  day  and 
a  year. 

TABLE. 

60  seconds  (s.)  are  1  minute m. 

60  minutes  "     1  hour h. 

@4  hours  "     1  day d. 

365  days  "     1  common  year .  c.  y. 

866  days  1  leap  year l*y* 

lc.y.  =  865d.  =  8760h.=52B600m.  =  31586000s. 

Also,  7  days  are  1  week,  12  months  are  1  year,  and 
100  years  are  1  century. 


DENOMINATE  NUMBERS.  149 

325.  The  Names  of  the  Months,  and  the  number  of 
days  in  each,  are — 


Days. 

Days. 

January,   1st  month, 

31. 

July, 

7th  month,  31. 

February,  2d     " 

28 

or  29. 

August, 

8th    "         31. 

March,      3d     " 

31. 

September, 

9th     "        30. 

April,       4th    " 

30. 

October, 

10th  "        31. 

May,         5th    " 

31. 

November, 

11th   "        30. 

June,         6th    " 

30. 

December, 

12th   "        31. 

The  exact  length  of  the  year  is  about  365^  days ;  hence,  every 
fourth  year  February  has  29  days,  and  the  year  has  366  days. 

In  general,  every  year  whose  number  can  be  divided  by  4 
without  a  remainder  is  leap  year. 

Thus,  the  year  1872  is  a  leap  year. 

326. — 1.  How  many  seconds  in  1  minute?  In  one 
half  a  minute  ?     In  one  third  of  a  minute  ? 

2.  How  many  minutes  in  120  seconds? 

3.  How  many  minutes  in  1  hour  ?     In  \  of  an  hour  ? 

4.  How  many  hours  in  120  minutes?  In  180 
minutes?     In  240  minutes? 

5.  How  many  hours  in  2  days  ?     In  3  days  ? 

6.  What  part  of  a  day  is  6  hours?     Is  18  hours? 

7.  How  many  days  in  48  hours?  What  part  of  a 
day  are  6  hours  ? 

8.  How  many  months  in  5  years  ?     In  9  years  ? 

9.  How  many  weeks  in  49  days  ?     In  63  days  ? 

10.  How  many  days  has  a  leap  year?  A  common 
year? 

11.  What  months  have  30  days  each  ?  In  what  years 
does  February  have  29  days  ? 

12.  How  many  days  are  there  in  7  weeks?  In  9 
weeks  ? 

13* 


150 


DENOMINATE   NUMBERS. 


WRITTEN  EXERCISES. 

327. — 10  How  many  minutes  in  18  days? 

2.  How  many  days  in  25920  minutes  ? 

3.  How  many  hours  in  a  common  year  ? 

4.  How  many  seconds  in  24  hours  ? 

5.  How  many  hours  in  86400  seconds  ? 

6.  How  many  years  in  8760  hours? 

7.  How  many  seconds  in  a  common  year  ? 

8.  How  many  days  in  31622400  seconds? 

9.  If  you  are  11  years  old,  how  many  minutes  have 
you  lived,  allowing  365^  days  to  a  year  ? 


SECTION   XXVIII. 


PAPER  AND  COUNTING. 


328.  Paper  is  bought 
and  sold  by  the  sheet, 
quire,  ream,  bundle  or 
bale. 

TABLE. 
24  sheets  are  1  quire. 
20  quires  "   1  ream. 
2  reams     "    1  bundle. 
5 bundles  "    lbale. 

329.  In  counting  cer- 
tain articles,  the  units 
dozen,  gross  and  great 
gross  are  used. 


DENOMINATE  NUMBERS.  151 

f 
TABLE. 

12  things  are  1  dozen doz. 

12  dozen      "    1  gross gro. 

12  gross       "    1  great  gross.  .  grt.  gro. 

Two  things  are  a,  pair,  and  20  things  are  a  score. 

330. — 1.  How  many  sheets  of  paper  in  2  quires? 

2.  How  many  quires  in  1  bundle  ?     In  2  bundles  ? 

3.  How  many  quires  in  48  sheets  ? 

4.  How  many  reams  in  80  sheets  ?     In  100  sheets  ? 

5.  How  many  things  in  5  dozen  ?     How  many  pens 
in  a  gross  ? 

6.  How  many  dozen  in  a  great  gross  ?     In  half  of  a 
great  gross  ? 

7.  How  many  pairs  are  40  ?     How  many  scores  are 
40?    Are  80?     Are  120? 

WRITTEN    EXERCISES. 

331. — 1.  How  many  sheets  are  9|  reams? 

2.  How  many  reams  are  4560  sheets  ? 

3.  How  much  will  a  gross  of  pens  cost,  at  16  cents  a 
dozen  ? 

4.  If  you  buy  5T3o  reams  of  paper,  how  many  sheets 
will  you  obtain  ? 

5.  In  some  boxes  there  are  2064  eggs ;  what  is  their 
value  at  30  cents  a  dozen  ? 

6.  How  many  boxes  will  be  required  to  pack  2592 
screws,  if  each  box  will  hold  9  dozen  ? 

7.  How  much  will  25  gross  of  pens  cost,  at  2  cents 
per  pen  ? 

8.  What  will  75  bundles  of  paper  cost,  at  15  cents 
per  quire  ? 


152  BE  VIEW. 

332.  Test  Questions. — 1.  What  is  a  circle?  A  circumfer- 
ence?   An  arc?     A  degree? 

2.  What  is  an  angle?     The  measure  of  an  angle? 

3.  What  are  circular  measures?  The  units  of  circular  meas- 
ures?   Kecite  the  table. 

4.  How  many  miles  is  a  minute  of  circumference? 

5.  What  are  measures  of  time?  The  units  of  measures  of 
time?     Recite  the  table. 

6.  How  many  days  are  1  week?  How  many  months  are  1 
year?     How  many  years  are  1  century? 

7.  Name  the  months,  and  give  the  number  of  days  in  each. 

8.  What  is  the  exact  length  of  a  year?  How  often  has  Feb- 
ruary 29  days  ?  When  February  has  29  days,  how  many  days 
has  the  year  ?     In  general,  what  years  are  leap  years  ? 

9.  By  what  is  paper  bought  and  sold?  Recite  the  table. 
What  units  are  used  in  counting  certain  articles  ?  Recite  the 
table.  How  many  things  are  a  pair  ?  How  many  things  are  a 
score  ? 


SECTION   XXIX. 
REVIEW  OF  DENOMINATE  NUMBERS. 

333. — 1 .  How  many  inches  in  f  of  a  yard  ? 

Solution. — 1  yard  is  3  feet;  f  of  3  feet  are  2  feet;  and  since 
1  foot  is  12  inches,  2  feet  are  2  times  12  inches,  or  24  inches. 
Hence,  f  of  a  yard  are  24  inches. 

2.  How  many  pints  in  f  of  a  bushel  ? 

3.  At  f  of  a  dollar  a  cord  foot,  what  will  i  of  a  cord 
of  wood  cost  ? 

4.  At  $.10  per  ounce,  what  will  1£  pounds  of  spice 
cost? 

5.  What  part  of  a  bushel  is  3  pecks  ? 


REVIEW.  153 

6.  At  2  dollars  a  bushel,  what  will  3  pecks  of  wheat 
cost? 

7.  James  is  108  months  old,  and  his  age  is  f  of  his 
brother's.     What  is  his  brother's  age  ? 

8.  At  16  cents  a  yard,  what  will  9  yards  of  cloth 
cost? 

9.  At  12  cents  a  peck,  how  many  bushels  of  oats  can 
be  bought  for  96  cents  ? 

10.  If  40  perches  of  land  cost  9  dollars,  what  will  an 
acre  cost? 

11.  If  i^  of  a  ton  of  hay  be  worth  3  dollars,  what  will 
30  hundred-weight  be  worth  ? 

WRITTEN    EXERCISES. 

334. — 1 .  How  many  minutes  in  a  leap  year  ? 

2.  A  walk  is  40  rods  long ;  what  will  it  cost  to  pave 
it,  at  25  cents  per  linear  foot  ? 

3.  Into  how  many  lots,  of  20  square  rods  each,  can  an 
acre  and  a  half  be  divided  ? 

4.  How  much  must  be  paid  for  25  barrels  of  beef,  at 
1 1  cents  a  pound  ? 

5.  At  11  cents  a  pound,  how  many  pounds  of  beef 
can  be  bought  for  $550  ? 

6.  For  how  much  will  a  hogshead  of  liquor  sell,  if 
retailed  at  10  cents  a  half  gill  ? 

7.  A  certain  town  is  36  square  miles  in  extent ;  how 
many  acres  does  it  contain  ? 

8.  How  many  bushels  in  a  load  of  corn,  which,  at 
86  cents  a  bushel,  costs  $34.40  ? 

9.  How  many  hogsheads  of  molasses,  of  63  gallons 
each,  worth  55  cents  a  gallon,  can  be  bought  for  $1386  ? 


154  COMPOUND  NUMBERS. 


SECTION   XXX. 

REDUCTION  OF  COMPOUND  NUMBERS. 

335. — 1.  How  many  inches  are  6  feet  5  inches  ? 

Solution. — Since  1  foot  is  12  inches,  6  feet  must  be  6  times  12 
inches,  or  72  inches ;  and  72  inches  plus  5  inches  are  77  inches. 
Hence,  6  feet  5  inches  are  77  inches. 

2.  How  many  cord  feet  in  5  cords  6  cord  feet  ? 

3.  Reduce  1  peck  3  quarts  1  pint  to  pints. 

4.  Reduce  2  tons  9  hundred-weight  to  hundred-weights. 

5.  How  many  feet  in  77  inches  ? 

Solution. — Since  12  inches  are  1  foot,  77  inches  must  be  as 
many  feet  as  12  inches  are  contained  times  in  77  inches,  which 
are  6  times,  with  a  remainder  of  5  inches.  Hence,  77  inches 
are  6  feet  5  inches. 

6.  How  many  cords  in  46  cord  feet? 

7.  Reduce  37  gills  to  quarts. 

8.  How  many  pecks  in  33  pints? 

9.  How  many  tons  are  49  hundred-weight? 

10.  What  will  2  gallons  3  quarts  of  vinegar  cost,  at 
5  cents  a  pint? 

11.  A  wall  is  4  yards  6  inches  long ;  what  is  its  length 
in  inches  ? 

DEFINITIONS. 

336.  A  Simple  Denominate  Number  is  a  number  ex- 
pressed in  units  of  only  one  denomination. 

Thus,  5  yards  is  a  simple  denominate  number. 

337.  A  Compound  Denominate  Number  is  a  number  ex- 
pressed in  units  of  more  than  one  denomination. 

Thus,  6  yards  7  inches  is  a  compound  denominate  number. 


COMPOUND   NUMBERS.  155 

338.  Reduction  of  Denominate  Numbers  is  the  process  of 
changing  them  from  one  denomination  to  another  without 
changing  their  value. 

339.  Principle. — Denominate  numbers  are  reduced  tc 
lower  denominations  by  multiplication,  and  to  higher  de- 
nominations by  division, 

REDUCTION   DESCENDING. 

340.  Reduction  Descending  is  the  process  of  changing  a 
number  to  an  equivalent  number  expressed  in  units  of  a 
lower  denomination. 

WRITTEN   EXERCISES. 

341. — 1.  How  many  pints  are  3gal.  2qt.  lpt.  ? 

Solution. — Since  1  gallon  is  4  quarts, 
J  gal.  J  qt.  1  pt.       3  gajlons  must  be  3  times  4  quartS)  0r  12 

•^  quarts ;  and  12  quarts  plus  2  quarts  are 

14  qt.  14  quarts. 

2  Since   1   quart  is   2   pints,  14  quarts 

— —  must  be  14  times  2  pints,  or  28  pints; 

~J  PL  and  28  pints  plus  1  pint  are  29  pints. 

Hence,  3gal.  2qt,  lpt.  are  29pt. 

2.  Reduce  4yd.  2ft.  7in.  to  inches. 

3.  Reduce  5  sq.  yd.  7  sq.  ft.  27  sq.  in.  to  square  inches. 

342.  Rule  for  Reduction  Descending.— Multiply  the  num- 
ber of  the  highest  denomination  given,  by  that 
**  umber  of  the  next  lower  which  equals  one  of  the 
higher,  and  to  the  product  add  the  number,  if  any, 
of  the  next  lower  denomination. 

Reduce  this  result  in  lihe  manner,  and  so  proceed 
until  the  given  number  is  reduced  to  the  required 
denomination. 


156  COMPOUND   NUMBERS. 

PROBLEMS. 

343.— 1.  Reduce  3°  4'  16"  to  seconds. 

2.  Reduce  3 A.  140P.  to  square  yards. 

3.  Reduce  lmi.  138rd.  4yd.  to  feet. 

4.  Reduce  100  cu.  yd.  20  cu.  ft.  100  cu.  in.  to  cubic 
inches. 

5.  Reduce  7bu.  3pk.  5qt.  to  quarts. 

6.  Reduce  9oz.  llpwt.  17gr.  to  grains. 

7.  Reduce  4T.  9cwt.  91b.  to  ounces. 

8.  Reduce  lhhd.  20gal.  3qt.  to  quarts. 

9.  How  much  must  be  paid  for  1  A.  40  sq.  rd.  of  land, 
at  $1.50  per  square  rod? 

10.  What  will  it  cost  to  make  lmi.  lOOrd.  of  fence, 
at  $1.20  a  rod? 

11.  How  much  must  be  paid  for  13  63  29  of  med- 
icine, at  5  cents  a  grain  ? 

12.  How  many  hours  has  a  man  lived  who  is  50y. 
125d.  old,  allowing  365|  days  to  the  year? 

REDUCTION   ASCENDING. 

344.  Reduction  Ascending  is  the  process  of  changing  a 
number  to  an  equivalent  number,  expressed  in  units  of  a 
higher  denomination. 

WRITTEN  EXERCISES. 

345. — 1.  How  many  gallons  are  29  pints? 

Solution. — Since  2  pints  are 
2  pt.)29  pt.  1  quart,   29  pints   must   be   as 

A  at  )1A  at.  1  vt  many  quarts  as  2  pints  are  con- 

— r — r —  tained  times  in  29  pints,  which 

3  gal.  4  qt.  are  14  timeSj  witj1  a  remainder 

89 pt.^3  gal.  2  qt.  1  pt.      of  1  pint.     Hence,  29  pints  are 

14  quarts  1  pint. 


COMPOUND  NUMBERS.  157 

Since  4  quarts  are  1  gallon,  14  quarts  must  be  as  many  gallons 
as  4  quarts  are  contained  times  in  14  quarts,  which  are  3  times, 
with  a  remainder  of  2  quarts.  Hence,  14  quarts  are  3  gallons 
2  quarts  ;  and  29  pints  are  3  gallons  2  quarts  1  pint. 

2.  Reduce  175  inches  to  yards. 

3.  Reduce  7515  square  inches  to  square  yards. 

4.  Reduce  37038  seconds  to  hours. 

346.  Rule  for  Reduction  Ascending.— Divide  the  given 
number  by  that  number  of  its  denomination, 
whieh  equals  one  of  the  next  higher,  and  write 
the  remainder,  if  any. 

Divide  the  quotient  in  like  manner,  and  so  con- 
tinue  until  the  given  number  is  changed  to  the  re- 
quired denomination. 

The  last  quotient,  with  the  remainders,  if  any, 
written  in  their  order  from  the  highest  to  the 
lowest,  will  be  the  required  number. 

Reductions  Descending  and  Ascending,  being  per- 
formed by  opposite  processes,  are  proofs  of  each  other. 

PROBLEMS. 

347. — 1.  Reduce  11056  seconds  to  degrees. 

2.  Reduce  18755  square  yards  to  acres. 

3.  Reduce  7569  feet  to  miles. 

4.  Reduce  4700260  cubic  inches  to  cubic  yards. 
I      5.  Reduce  275  quarts  to  bushels. 

6.  Reduce  4601  grains  to  ounces. 

7.  Reduce  142494  ounces  to  tons. 

8.  Reduce  335  quarts  to  hogsheads. 

9.  How  many  acres  of  land  will  $300  purchase,  at 
$1.50  per  square  rod  ? 

14 


158  COMPOUND    NUMBERS. 

10.  How  many  miles  of  fence,  at  $1.20  per  rod,  can 
be  built  for  $504  ? 

11.  How  many  ounces  of  medicine,  at  5  cents  a  grain, 
can  be  bought  for  $44  ? 

12.  How  many  years  has  a  man  lived  who  is  438300 
hours  old,  allowing  365^  days  to  a  year  ? 


348.  Test  Questions.— 1.  What  is  a  denominate  number? 
A  simple  denominate  number?  A  compound  denominate  num- 
ber? 

2.  What  is  reduction  ?     The  principle  of  reduction  ? 

3.  What  is  reduction  descending  ?  How  is  a  number  reduced 
from  a  higher  to  a  lower  denomination  ? 

4.  What  is  reduction  ascending?  How  is  a  number  reduced 
from  a  lower  to  a  higher  denomination  ? 

5.  Why  are  reductions  descending  and  ascending  proofs  of 
each  other? 


SECTION    XXXI. 
ADDITION  OF  COMPOUND  NUMBERS. 

349. — 1.  What  is  the  sum  of  6  yards  1  foot,  and  4 
yards  1  foot? 

2.  Add  6  cords  3  cord  feet,  and  5  cords  6  cord  feet. 

Solution. — 6  cords  3  cord  feet,  and  6  cord  feet,  are  6  cords  9 
cord  feet,  or  7  cords  1  cord  foot ;  7  cords  1  cord  foot,  and  5  cords, 
are  12  cords  and  1  cord  foot. 

3.  Add  10  bushels  2  pecks,  and  4  bushels  3  pecks. 

4.  If  you  should  buy  in  one  month,  4  gallons  3  quarts 
of  kerosene,  and  the  next  month,  3  gallons  3  quarts,  how 
much  would  you  buy  in  all  ? 


COMPOUND  NUMBERS.  169 

5.  Mary  is  9  years  7  months  old,  and  Alice  is  8  years 
8  months  old.     What  is  the  sum  of  their  ages  ? 

6.  A  farmer  sold  two  turkeys  ;  one  weighed  10  pounds 
11  ounces,  and  the  other  11  pounds  7  ounces.  What 
was  the  weight  of  both  ? 

DEFINITION. 

350.  Addition  of  Compound  Numbers  is  the  process  of 
uniting  two  or  more  compound  numbers  of  the  same 
kind  to  find  their  sum. 

WRITTEN   EXERCISES. 

351_1.  What  is  the  sum  of  151b.  5oz.  15pwt.;  71b. 
9oz.  12pwt. ;  and  6oz.  4pwt.  ? 

Solution. — Write  the  numbers  so 
15  lb.   5  oz.  15  pwt.      that  figures   expressing   units   of  the 
Y         g         12  same  denomination  shall   be  in  the 

s>  v  same  columns. 

Begin  with  the  pennyweights,  and 


23  lb.  U  oz.  11  pwt.  add  the  pennyweights,  ounces  and 
pounds  separately. 

The  sum  of  the  pennyweights  is  31  pennyweights,  or  1  ounce 
11  pennyweights.  Write  the  11  pennyweights  under  the  column 
of  pennyweights,  and  reserve  the  1  ounce  to  add  with  the 
ounces. 

The  sum  of  the  ounces  is  21  ounces,  or  1  pound  9  ounces. 
Write  the  9  ounces  under  the  column  of  ounces,  and  reserve  the 
1  pound  to  add  with  the  pounds. 

The  sum  of  the  pounds  is  23  pounds,  which  write  under  the 
column  of  pounds. 

The  entire  sum  is  23  pounds  9  ounces  11  pennyweights. 

2.  What  is  the  sum  of  6mi.  120rd.  3yd. ;  7mi.  160rd. 
5yd. ;  and  55rd.  3yd.  ? 

3.  What  is  the  sum  of  15gal.  3qt.  lpt. ;  7gal.  0  qt. 
lpt. ;  and  2qt.  Opt.  ? 


160  COMPOUND  NUMBERS. 

352.  Rule  for  Addition  of  Compound  Numbers.—  Write  the 
numbers  so  that  units  of  the  same  denomination 
shall  stand  in  the  same  column. 

Begin  with  the  lowest  denomination,  and  add 
the  numbers  of  each  denomination  separately .  If 
the  sum  is  less  than  one  of  the  next  higher  denomi- 
nation, write  it  as  a  part  of  the  required  result. 

If  the  sum  is  equal  to  or  exceeds  one  of  the  nexv 
higher  denomination,  reduce  it  to  that  denomi- 
nation, wi'ite  the  remainder,  if  any,  as  a  part  of 
the  required  result,  and  add  the  units  of  the  higher 
denomination  with  the  column  of  that  denomi- 
nation. 

r  B  OK  L  EMS. 

353.  Write  and  add— 

(1.)  (2.) 

15cwt.   75  lb.   7  oz.  10  A.    146  sq.rd. 

3           0       5  73          15 

25       9  3          75 


(3.) 

(4.) 

5  CM. 

.yd.  17  cu.ft.   703  eu.in. 

9  oz.    18  pwt.   11  gr. 

11 

10            835 

1        19          23 

11            106 

4         0         20 

3 

9            112 

6         0            0 

(5.) 

(6.) 

100  bu.  3  ph. 

25  rd.  3  yd.  2  ft. 

76        2 

16        £        1 

13        3 

11        3        0 

3 

4       2 

COMPOUND  NUMBERS.  161 

7.  What  is  the  sum  of  2d.  7h.  6m. ;  5d.  13h.  25m. ; 
llh.  11m.;  and  7d.  15h.  55m.? 

8.  Add  1°  30'  15"     19°  45'  17"  ;  and  31°  40'  16". 

9.  A  boy  gathered  in  one  day,  lpk.  3qt.  lpt.  of  ber- 
ries; another  day,  lpk.  5qt.  lpt. ;  and  a  third  day,  2pk. 
2qt.  lpt.     How  many  berries  did  he  gather  in  all  ? 

10.  In  one  field  there  are  17 A.  120  sq.  rd. ;  and  in 
another  15A.  140  sq.  rd.  How  much  land  is  there  in 
both  fields? 

11.  In  one  car  there  are  16T.  llcwt.  751b.  of  coal; 
in  another,  15T.  19cwt.  311b. ;  and  in  a  third,  14T. 
17cwt.  501b.  How  much  coal  is  there  in  the  three 
cars? 


SECTION   XXXII. 

SUBTRACTION  OF   COMPOUND    NUMBERS. 

354. — 1.  In  one  sack  there  are  2bu.  3pk.  of  wheat, 
and  in  another,  lbu.  2pk.  How  much  more  is  there  in 
one  sack  than  in  the  other  ? 

2.  If  5ft.  lOin.  be  cut  from  a  line  7ft.  9in.  long,  how 
much  of  the  line  will  be  left  ? 

Solution. — There  will  be  left  of  the  line  the  difference  be- 
tween 7  feet  9  inches  and  5  feet  10  inches.  7  feet  9  inches  les9 
10  inches  are  6  feet  11  inches ;  and  6  feet  11  inches  less  5  feet 
are  1  foot  11  inches.     Hence,  there  will  be  left  1  foot  11  inches. 

3.  If  you  should  have  in  a  cask  llgal.  3qt.  of  molasses, 
and  should  sell  8gal.  3qt.  of  it,  how  much  would  there 
be  left? 

4.  From  lOh.  30m.  subtract  8h.  40m. 


162  COMPOUND  NUMBERS. 

5.  In  one  firkin  there  are  121b.  8oz.  of  butter,  and 
in  another  91b.  12oz.  How  much  more  is  there  in  the 
one  than  in  the  other  ? 

6.  In  one  load  there  are  2cd.  1  cd.  ft„  of  wood,  and 
in  another  load  led.  7  cd.  ft.  How  much  is  required  to 
make  the  smaller  load  equal  the  greater  ? 

DEFINITION. 

355.  Subtraction  of  Compound  Numbers  is  the  process  of 
finding  the  difference  between  two  compound  numbers 
of  the  same  kind. 

WRITTEN  EXERCISES. 

356.— 1.  From  17bu.  2'pk.  6qt.  take  8bu.  3pk.  4qt. 

17  h      2    k    ft    t  Solution.  —  Write   the    subtrahend 

Q  .  under  the  minuend,  so  that  units  of  the 

M 2 *^  same   kind    shall    stand    in    the    same 

8  bu.  3pk  2  ql      columns. 

Begin  at  the  right  and  subtract  the 
units  of  each  denomination  of  the  subtrahend  from  those  of  the 
same  denomination  in  the  minuend. 

4  quarts  from  6  quarts  leaves  2  quarts,  which  is  the  difference 
of  the  quarts. 

Since  3  pecks  cannot  be  taken  from  2  pecks,  take  1  bushel 
from  the  17  bushels,  leaving  16  bushels,  and  add  it,  reduced,  to 
the  2  pecks,  thus  obtaining  6  pecks ;  then,  3  pecks  from  6  pecks 
leaves  3  pecks,  which  is  the  difference  of  the  pecks. 

8  bushels  from  16  bushels  leaves  8  bushels,  which  is  the  dif- 
ference of  the  bushels. 

The  entire  difference  is  8  bushels  3  pecks  2  quarts. 

2.  From  131b.  5oz.  16pwt.  subtract  91b.  7oz.  14pwt. 

3.  From  6gal.  2qt.  Opt.  subtract  4gal.  3qt.  lpt. 

4.  From  22yd.  2qr.  2na.  subtract  13yd.  3qr.  3na. 


COMPOUND  NUMBERS.  163 

357.  Rule  for  Subtraction  of  Compound  Numbers.—  Write 
the  subtrahend  under  the  minuend,  so  that  units 
of  the  same  denomination  shall  be  in  the  same 
column. 

Begin  with  the  lowest  denomination,  and  subtract 
the  units  of  each  denomination  of  the  subtrahend 
from  those  of  the  same  denomination  in  the  minu- 
end, if  possible,  and  write  the  difference  beneath. 

If  the  number  of  any  denomination  of  the  sub- 
trahend is  greater  than  that  of  the  same  denomi- 
nation in  the  minuend,  increase  the  number  in  the 
minuend,  by  adding  to  it  as  many  units  as  make 
one  of  the  next  higher  denomination,  and  subtract; 
then,  regarding  the  number  of  the  next  higher  de- 
nomination of  the  minuend  as  one  less,  proceed  as 
before. 

PROBLEMS. 

358.  Write  and  subtract — 

(1.)  (2.) 

From  25  mi.   100  rd.  7  yd.  9  A.  120  P.  24  sq.  yd. 

Take    16         120        6  6        125       19 


From 
Take 

(3.) 
15  wk.  0  d.   10  h. 
12         9       11 

From 
Take 

(5.) 

11 1  0S  14  3 

7      2      16 

(4.) 

44  T. 

75  lb. 

10  oz. 

21 

80 

15 

(6.) 

3° 

7' 

0" 

1 

15  . 

45 

7.  A  man  who  has  started  on  a  journey  of  63mi. 
160rd.  has  traveled  44mi.  125rd.  3yd.  How  much 
farther  has  he  to  travel  to  complete  his  journey  ? 


164  COMPOUND   NUMBERS. 

8.  James  is  13  years  9  months  old,  and  Henry  is  11 
years  11  months  old.     What  is  the  difference  in  their 

i? 


9.  From  a  cask  which  contained  75  gallons  of  vinegar 
there  has  been  drawn  46gal.  3qt.  lpt.  How  much  re- 
mains in  the  cask  ? 

10.  The  longitude  of  Boston  is  71°  3'  30"  West,  and 
of  Chicago,  87°  37'  47"  West.  What  is  the  difference 
of  their  longitudes  ? 

11.  What  is  the  time  from  May  17, 1869,  to  June  16, 
1871? 

Solution.— The  time  from  May 
1871  y.  6  mo.   16  d.     17>  1869j  to  May  17>  m^  ig  2^ .  from 

1869       5  17  May  .17,  1871,  to  June  16,  1871,  is 

2  y.   0  mo.   30  d.       30d.     The  entire  difference  of  time 
is  2y.  30d. 

12.  A  man  was  born  May  16,  1819;  how  old  was 
he  July  4,  1871  ? 


359.  Test  Questions. — 1.  What  is  addition  of  compound 
numbers  ?     How  do  you  write  the  numbers  for  adding  ? 

2.  How  do  you  add  the  units  of  the  several  denominations  ? 
How  do  you  proceed  if  the  sum  of  the  units  of  any  denomina- 
tion is  less  than  a  unit  of  the  next  higher  denomination?  How 
do  you  proceed  if  their  sum  is  equal  to  or  greater  than  a  unit 
of  the  next  higher  denomination? 

3.  What  is  subtraction  of  compound  numbers?  How  do  you 
write  the  numbers  for  subtracting  ? 

4.  How  do  you  subtract?  How  do  you  proceed  if  the  num- 
ber of  any  denomination  of  the  subtrahend  is  greater  than  that 
above  it  ? 


COMPOUND  NUMBERS.  165 


SECTION    XXXIII. 

MULTIPLICATION  OF  COMPOUND 
NUMBERS. 

360.— 1.  If  a  box  contain  2  hundred-weight  20 
pounds  of  sugar,  how  much  will  2  similar  boxes  contain  ? 

2.  How  much  wood  is  there  in  3  ranges,  each  con- 
taining 3  cords  5  cord  feet  ? 

Solution. — 3  times  5  cord  feet  are  15  cord  feet,  or  1  cord  7 
cord  feet,  and  3  times  3  cords  are  9  cords ;  9  cords  and  1  cord  7 
cord  feet  are  10  cords  7  cord  feet,  the  answer  required. 

3.  John  is  9  years  4  months  old,  and  his  father  is  4 
times  as  old.     How  old  is  his  father  ? 

4.  A  boy  gathered  1  peck  3  quarts  of  berries  every 
day  for  6  days.     How  many  did  he  gather  in  all  ? 

5.  If  it  take  1  ounce  3  drams  of  medicine  for  a  pre- 
scription, how  much  will  it  take  for  6  similar  prescrip- 
tions ? 

6.  If  4  yards  3  quarters  of  cloth  are  required  for  a 
suit  of  clothes,  how  many  yards  will  be  required  for  7 
suits? 

7.  How  much  wood  is  there  in  two  ranges,  each  con- 
taining 2  cords  6  cord  feet? 

8.  If  a  team  can  plow  1  acre  40  square  rods  in  1  day, 
how  much  can  it  plow  in  4  days  ? 

DEFINITIONS. 

361.  Compound  Multiplication  is  the  process  of  taking 
a  compound  number  as  many  times  as  there  are  units  in 
the  multiplier. 


166  COMPOUND  NUMBERS. 

WRITTEN  EXERCISES. 

362.— 1.  Multiply  6gal.  3qt.  lpt.  by  5. 

6  gal  3  qt.  1  pL  Solution. — Write  the  multiplier  urn 

q  der    the    lowest    denomination    of   the 

■— —  multiplicand. 

34  gal  1  qt.  1  pt.  Begin  at  the  right  and  muitiply  the 

number  of  each  denomination  in  the  order  of  the  denominations. 

5  times  1  pint  are  5  pints,  or  2  quarts  1  pint ;  write  the  1  pint 
as  the  number  of  that  denomination  in  the  product,  and  reserve 
the  2  quarts  to  be  added  to  the  product  of  the  quarts. 

5  times  3  quarts  are  15  quarts;  15  quarts  and  the  2  quarts  re- 
served are  17  quarts,  or  4  gallons  1  quart.  Write  the  1  quart 
as  the  number  of  that  denomination  in  the  product,  and  reserve 
the  4  gallons  to  be  added  to  the  product  of  the  gallons. 

5  times  6  gallons  are  30  gallons ;  30  gallons  and  the  4  gallons 
reserved  are  34  gallons,  which  write  as  the  gallons  of  the  product. 

The  entire  product  is  34  gallons  1  quart  1  pint. 

2.  Multiply  6rd.  2yd.  1ft.  by  5. 

3.  Multiply  101b.  3pwt.  5gr.  by  8. 

363.  Rule  for  Multiplication  of  Compound  Numbers.—  Write 
the  multiplier  under  the  lowest  denomination  of  the 
multiplicand. 

Begin  with  the  lowest  denomination,  and  multi- 
ply the  number  of  each  denomination  in  its  order. 

If  the  product  is  less  than  one  of  the  next  higher 
denomination,  write  it  as  a  part  of  the  required 
product. 

If  the  product  is  equal  to  or  exceeds  one  of  the 
next  higher  denomination ,  reduce  it  to  that  denom- 
ination, write  the  remainder ,  if  any,  as  a  part  of 
the  required,  product,  and  add  the  units  of  the 
higher  denomination  to  the  product  of  that  denom- 
ination. 


COMPOUND  NUMBERS.  167 

PROBLEMS. 

364.  Copy  and  multiply — 

(1.)                                                   (2.) 
1  hhd.  10  gal  3  qt.                 50  eu.ft.  1121  m.  in. 
7  8 

(3.)  (4.) 

U  A.  63  P.  1  sq.  yd.  4  bu.  3  pk.  2  qt. 

4  9 


(5.)  (6.) 

6  wk.  3  d.  10  h.  13°  a  U" 

10  8 


7.  A  farmer  sold  6  loads  of  wood,  each  containing 
2cd.  5  cd.  ft.     How  much  did  he  sell  in  all  ? 

8.  What  is  the  weight  of  7  boxes  of  sugar,  if  each 
weighs  lcwt.  311b.  8oz.? 

9.  Jane  is  8y.  3mo.  old ;  how  old  is  her  aunt,  who  is 
5  times  as  old  ? 

10.  If  a  train  of  cars  move  27mi.  120rd.  2yd.  in  an 
hour,  how  far  will  it  move  at  the  same  rate  in  11  hours? 

11.  Flint  has  in  his  farm  40 A.  35P.,  and  my  farm  is 
7  times  as  large.     What  is  the  extent  of  my  farm  ? 

12.  How  much  molasses  is  there  in  12  casks,  each 
containing  62gal.  3qt.  lpt.  ? 

13.  If  it  require  loz.  lOpwt.  15gr.  of  silver  to  make 
1  spoon,  how  much  will  be  required  to  make  18  spoons? 

14.  In  a  certain  to.wn  there  are  3  farms.  Each  farm 
is  divided  into  9  equal  lots,  and  each  lot  contains  12 A. 
72P.     What  is  the  extent  of  the  3  farms  ? 


168  COMPOUND   NUMBERS, 

SECTION    XXXIV. 

DIVISION  OF  COMPOUND  NUMBERS. 

365. — 1.  If  6  pounds  12  ounces  of  tea  be  divided 
equally  among  3  persons,  how  much  will  each  receive  ? 

2.  If  8    bushels   3   pecks   of  chestnuts   be   divided 
equally  among  5  boys,  how  many  will  each  boy  receive? 

Solution.  —  Each  boy  will  receive  one  fifth  of  8  bushels 
3  pecks.    One  fifth  of  8  bushels  is  1  bushel,  with  a  remainder  of 

3  bushels ;  3  bushels  are  12  pecks,  and  12  pecks  plus  3  pecks 
are  15  pecks ;  one  fifth  of  15  pecks  is  3  pecks.  Hence,  each  boy 
will  receive  1  bushel  3  pecks. 

3.  Johnson  is  37  years  4  months  old,  and  his  age  is 

4  times  that  of  his  son.     What  is  the  age  of  his  son  ? 

4.  If  it  require  33  yards  1  quarter  to  make  7  suits, 
how  much  will  it  require  to  make  1  suit  ? 

5.  In  3  equal  ranges  of  wood  there  are  10  cords  7 
cord  feet.     How  much  is  there  in  each  range  ? 

DEFINITION. 
366.  Division  of  Compound  Numbers  is  the  process  of 
finding  how  many  times  one  denominate  number  is  con- 
tained in  another  of  a  similar  kind ;  or,  it  is  the  process 
of  separating  a  compound  number  into  equal  parts. 

WRITTEN   EXERCISES. 

367.-1.  Divide  34gal.  lqt.  lpt.  by  5. 

5)34  gal  1  gt.  1  pt.  Solution.— Write  the  divisor  at 

~       z~z    "    ^T        the  left  of  the  dividend. 

0  gal.  d  qt.  1  pt.  Begin  with  the  highest  denomina- 

tion, and  divide  the  number  of  each  denomination  in  its  order. 
One  fifth  of  34  gallons  is  6  gallons,  with  a  remainder  of  4 


COMPOUND   NUMBERS.  169 

gallons.  Write  the  6  gallons  as  the  gallons  of  the  quotient ;  the  4 
gallons  are  16  quarts,  which,  added  to  the  1  quart,  give  17  quarts. 

One  fifth  of  17  quarts  is  3  quarts,  with  a  remainder  of  2 
quarts.  Write  the  3  quarts  as  the  quarts  of  the  quotient ;  the  2 
quarts  are  4  pints,  which,  added  to  the  1  pint,  give  5  pints. 

One  fifth  of  5  pints  is  1  pint,  which  write  as  a  part  of  the 
quotient. 

The  entire  quotient  is  6  gallons  3  quarts  1  pint. 

2.  Divide  32rd.  Oyd.  1ft.  by  5. 

3.  Divide  811b.  5pwt.  16gr.  by  8. 

368.  Rule  for  Division  of  Compound  Numbers.— Begin  with 
the  highest  denomination;  divide  the  number  of 
eaeh  denomination  in  its  order,  and  write  the 
several  quotients  as  the  parts  of  the  same  denom- 
inations of  the  required  quotient. 

When  there  is  a  remainder  reduce  it  to  the  next 
lower  denomination,  and  add  the  result  to  the 
number  of  that  denomination  before  dividing. 

When  divisor  and  dividend  are  both  compound 
numbers,  they  must  be  reduced  to  simple  denom- 
inate numbers  of  the  same  denomination  prepar- 
atory to  dividing. 

Multiplication  of  compound  numbers  and  division  of 
compound  numbers,  being  performed  by  opposite  pro- 
cesses, are  proofs  of  each  other. 

PROBLEMS. 

369.  Copy  and  divide — 

(1.)  (2.) 

7)7  hhd.  4.5  gal.  1  qt.  8)4-05  cu.  ft.  328  cu.  in, 

(3.)  (4.) 

4)U  A.  63  P.  lsq.yd.  9)37  bu^J  ph.  2  qt 

15 


170  COMPOUND  NUMBERS. 

5.  If  6  persons  should  share  equally  19cd.  6  cd.  ft. 
of  wood,  how  much  would  each  person  receive  ? 

6.  Mary's  mother  is  40y.  3mo.  old,  and  Mary  is  one 
fifth  as  old.     How  old  is  Mary  ? 

7.  If  7  equal  boxes  of  sugar  weigh  7cwt.  271b.  8oz., 
what  is  the  weight  of  one  box  ? 

8.  If  a  train  of  cars  move,  at  a  uniform  rate,  308m. 
44rd.  in  11  hours,  how  far  does  it  move  in  1  hour? 

9.  My  farm  contains  281A.  85P.,  and  Flint's  farm 
is  one  seventh  as  large.     How  large  is  Flint's  farm  ? 

10.  When  22oz.  19pwt.  lOgr.  of  silver  are  required 
to  make  18  spoons,  what  is  the  weight  of  1  spoon? 

11.  A  farmer  has  65bu.  2pk.  4qt.  of  grain,  which 
he  wishes  to  put  in  bags,  each  containing  lbu.  3pk.  4qt. 
How  many  bags  will  be  required  ? 

12.  In  a  certain  town  there  are  3  farms,  and  each  is 
divided  into  9  equal  lots.  The  entire  extent  of  the  farms 
is  37 A.  56P. ;  what  is  the  size  of  each  lot  ? 

13.  In  what  time  can  a  man  saw  1  cord  of  wood,  if 
he  can  saw  12  cords  in  93h.  lOmin.? 


370.  Test  Questions. — 1.  What  is  multiplication  of  com- 
pound numbers?  How  do  you  write  the  numbers  for  multiply- 
ing ?     How  do  you  multiply  ? 

2.  How  do  you  proceed  if  any  product  is  less  than  one  of  the 
next  higher  denomination  ?  If  any  product  is  equal  to  one  or 
more  units  of  the  next  higher  denomination? 

3.  What  is  division  of  compound  numbers?  How  do  you 
divide?     How  do  you  proceed  if  there  is  a  partial  remainder? 

4.  Why  are  the  multiplication  of  compound  numbers  and 
the  division  of  compound  numbers  proofs  of  each  other? 


REVIEW.  Ill 

SECTION    XXXV. 

BE  VIEW  OF  COMPOUND  NUMBERS. 

371. — 1.  When  oats  are  10  cents  a  peck,  what  will  1 
bushel  2  pecks  4  quarts  of  oats  cost  ? 

2.  Reduce  f  of  a  yard  to  units  of  lower  denom- 
inations. 

Solution. — Since  1  yard  is  3  feet,  f-  of  a  yard  is  f  of  3  feet, 
which  is  V5  of  a  *oot>  or  2  ^eet  anc*  §>  or  h  °f  a  foot.  Since  1 
foot  is  12  inches,  J  of  a  foot,  is  \  of  12  inches,  or  6  inches. 
Hence,  £  of  a  yard,  reduced  to  units  of  lower  denominations, 
is  2  feet  6  inches. 

3.  Reduce  f  of  an  acre  to  units  of  lower  denominations. 

4.  Reduce  f  of  a  ton  to  units  of  lower  denominations. 

5.  2  gallons  3  quarts  are  how  many  pints  ? 

6.  Reduce  31  pints  to  units  of  higher  denominations. 

Solution. — 31  pints  are  15  quarts  1  pint,  and  15  quarts  are  3 
gallons  3  quarts.  Hence,  31  pints  in  units  of  higher  denomina- 
tions are  3  gallons  3  quarts  1  pint. 

7.  Reduce  39  inches  to  units  of  higher  denominations. 

8.  Reduce  46  ounces  Troy  to  units  of  higher  denom- 
inations. 

9.  How  many  hours  from  9  o'clock  A.  m.  to  2  o'clock 

P.  M.  ? 

10.  How  many  minutes  from  15  minutes  before  10 
o'clock  to  15  minutes  past  11  o'clock? 

11.  James  has  lib.  8oz.  of  candy,  and  Edwin  has  f 
as  much.     How  much  has  Edwin  ? 

Solution. — 1  pound  8  ounces  is  24  ounces.  Hence,  James 
has  24  ounces,  and  since  Edwin  has  f  as  much,  he  has  f  of  24 
ounces,  or  18  ounces,  which  is  1  pound  2  ounces. 


172  REVIEW. 

12.  If  you  have  1  pound  9  ounces  of  gold,  and  should 
sell  f  of  it,  how  much  would  you  have  left  ? 

13.  At  $2  a  hundred- weight,  what  will  |  of  2  tons  5 
hundred- weight  of  hay  cost  ? 

14.  If  you  should  gather  in  one  day  3  quarts  1  pint 
of  berries,  and  the  next  day  5  quarts  1  pint,  how  much 
would  they  be  worth,  at  12  cents  a  quart? 

15.  How  many  months  old  is  a  boy  who  has  lived  11 
years  less  8  months  ? 

16.  Jason  has  1  pound  6  ounces  of  candy,  and  his 
brother  has  3  times  as  much.  How  much  more  than 
Jason  has  his  brother  ? 

17.  My  granary  has  two  bins;  one  will  hold  36 
bushels  1  peck,  and  the  other  one  fifth  as  much.  How 
much  less  does  one  contain  than  the  other  ? 

WRITTEN  EXERCISES. 

372. — 1.  How  many  steps,  of  3  feet  each,  must  you 
take  in  walking  1  mile  6  rods  ? 

2.  How  many  miles  has  a  man  travelled  who  has  gone 
1793  steps  of  3  feet  each? 

3.  I  have  2  plots  of  ground,  each  containing  1A. 
120P.  Into  how  many  lots,  of  20  square  rods  each,  can 
the  plots  be  cut  ? 

4.  A  planet  has  moved  in  a  given  time  17°  30'. 
How  many  seconds  has  it  moved  ? 

5.  At  90  cents  a  cord  foot,  how  many  cords  of  wood 
can  be  bought  for  $16.20? 

6.  I  have  2  pounds  of  silver.  How  many  spoons, 
weighing  loz.  9pwt.  each,  can  be  made  from  it,  and  how 
many  pennyweights  will  remain  ? 


REVIEW.  173 

7.  How  many  days,  of  8  hours  each,  do  you  waste  in 
a  term  of  12  weeks,  each  week  containing  5  school  days, 
if  you  are  idle  2  hours  each  school  day  ? 

8.  For  how  much  will  4bu.  2pk.  of  strawberries  sell, 
if  put  into  quart  boxes  and  retailed  at  30  cents  a  box  ? 

9.  What  will  4  casks  of  molasses,  each  containing 
84gal.  3qt.,  sell  for,  at  15  cents  a  quart? 

10.  How  many  loads,  each  containing  led.  2  cd.  ft., 
can  be  taken  from  a  range  of  wood  containing  25cd. 
6cd.  ft.  ? 

11.  If  a  train  of  cars  can  move  in  1  hour,  25mi.  80rcL, 
in  what  time  can  it  move  250mi.  160rd.  ? 

12.  How  many  kegs,  each  containing  5gal.  3qt.  lpt., 
can  be  filled  from  a  cask  containing  58gal.  3qt.  ? 


373.  Test  Questions.— 1.  What  is  a  denomination?  What 
are  the  denominations  of  linear  measure?  Of  surface  measure? 
Of  cubic  measure? 

2.  What  are  the  measures  of  lines,  surfaces  or  solids  called  ? 
Measures  of  extension.  Recite  the  table  of  linear  measures. 
The  table  of  surface  measures.     The  table  of  cubic  measures. 

3.  What  are  the  denominations  of  liquid  measure?  Of  dry 
measure?  What  are  the  measures  of  liquids,  or  of  grain,  fruits 
and  like  articles,  called?  Measures  of  capacity.  Recite  the 
table  of  liquid  measures.     Of  dry  measures. 

4.  What  are  the  denominations  of  Avoirdupois  Weight  ?  Of 
Troy  Weight?  What  are  these  called?  Measures  of  weight 
Recite  the  table  of  Avoirdupois  Weight.     Of  Troy  Weight. 

5.  What  are  the  denominations  of  circular  measure  ?  Recite 
the  table.  What  are  the  denominations  of  the  measure  of  time? 
Recite  the  table. 

6.  How  does  reduction  descending  differ  from  reduction 
ascending?  Compound  addition  from  compound  subtraction? 
Compound  multiplication  from  compound  division? 

15* 


174 


DECIMALS. 


SECTION    XXXVI. 

NOTATION  AND    NUMERATION    OF 

DECIMALS. 


374.— 1.  If  a  block  of  wood  be  divided  into  ten  equal 
parts,  what  is  1  of  the  parts  called  ?  What  are  2  of 
the  parts  called  ?     3  of  the  parts  ?     7  of  the  parts  ? 

2.  If  one  tenth  of  a  block  of  wood  be  divided  into 
ten  equal  parts,  what  is  1  of  the  parts  called  ?  What  are 
2  of  the  parts  called  ?     3  of  the  parts  ?     7  of  the  parts  ? 

3.  What  part  of  1  is  A  of  A?  A  of  A?  A  of  A? 
A  of  A? 

4.  What  part  of  one  tenth  is  one  hundredth  ?  How 
many  hundredths  is  one  tenth  ? 


DECIMALS.  175 

5.  If  one  hundredth  of  a  block  of  wood  be  divided 
into  ten  equal  parts,  what  is  1  of  the  parts  called  ? 
What  are  2  of  the  parts  called  ?  3  of  the  parts  ?  7 
of  the  parts  ? 

6.  What  part  of  1  is  TV  of  ^  of  TV  ?     fV  of  TV  of  -^  ? 

7.  What  part  of  one  hundredth  is  one  thousandth  ? 
How  many  thousandths  is  one  hundredth  ? 

DEFINITIONS. 

375.  A  Decimal  Fraction,  or  Decimal,  is  a  number  of 
tenths,  hundredths,  thousandths,  etc.,  expressed  by  placing 
a  point  (.)  before  the  numerator  and  omitting  the  de- 
nominator. 

Thus,  .3  expresses  3  tenths,  or  { q  ;  and  .03  expresses  3  hun- 
dredths, or  t§q. 

376.  The  point  is  called  the  Decimal  Point,  and  in  a 
numerical  expression  separates  the  decimal  from  the 
order  of  ones  or  unite. 

Thus,  3.003  expresses  3  ones  and  3  thousandths,  or  3To3oq. 

377.  The  first  order  at  the  right  of  the  decimal  point 
is  tenths,  the  second  hundredths,  the  third  thousandths?  etc., 
as  shown  in  the  following 


TABLE. 


I 


•a  -a 

I    i  •  I     I 

654321      .23456     7 


INTEGERS.  DECIMALS. 


176  DECIMALS. 

A  Mixed  Decimal  consists  of  an  integer  and  a  decimal. 

Thus,  15.165,  which  is  read  fifteen  ones  and  one  hundred  sixty- 
five  thousandths,  or  fifteen  and  one  hundred  sixty-fice  thousandths, 
is  a  mixed  decimal. 

378.  Principles. — 1.  Ten  of  any  decimal  order  are 
always  equal  to  one  of  the  next  higher  order. 

2.  The  denominator  of  a  decimal  is  1,  with  as  many 
ciphers  annexed  as  there  are  orders  in  the  decimal. 

WRITTEN  EXERCISES. 

379—1.  Write  and  read  .308. 

Solution. — .308  =  3  tenths  0  hundredths  8  thousandths;  or, 
.3  =  30  hundredths  a*  300  thousandths  ;  and  300  thousandths 
-f-  8  thousandths  =  308  thousandths.  Hence,  .308  is  read 
S08  thousandths. 

2.  Write  and  read  61.25. 

Solution. — 61.25  =  61  ones  and  decimal  .25 ;  .25  =  2  tenths 
5  hundredths ;  or,  since  .2  =  20  hundredths,  .25  =  20  hundreths 
-f  5  hundredths  =  25  hundredths.  Hence,  61.25  is  read  61  ones 
and  25  hundredths ;  or  61  and  25  hundredths. 

3.  Write  and  read  .3165. 

4.  Write  and  read  515.006. 

5.  Write  three  hundred  five  thousandths. 

Solution. — 305  thousandths  =  30  hundredths  and  5  thou- 
sandths ;  30  hundredths  =  3  tenths  and  0  hundredths.  Hence, 
305  thousandths  ==  3  tenths  0  hundredths  5  thousandths  =  .305. 

6.  Write  one  thousand  three  hundred  thirty-one  ten- 
thousandths. 

7.  Write  eleven  thousand  eleven,  and  eleven  thou- 
sandths. 


DECIMALS. 


177 


380.  Rules  for  Reading  and  Writing  Decimals.- -Read  a 
decimal  as  though  it  were  an  integer,  giving  it 
the  name  of  its  right-hand  order. 

Write  a  decimal  as  though  it  were  an  integer, 
and  place  the  decimal  point  so  that  each  figure 
shall  stand  in  its  proper  order,  marking  the  ab- 
sence of  units  of  any  order  by  a  cipher. 


PROBLEMS. 

381.  Write  and  read — 


1.  .347 

4.  .138 

7.  14.005 

2.  .1405 

5.  .10065 

8.  9.093 

3.  .0072 

6.  .000002 

9.  1.6444 

Write  in  the  decimal  form — 

io.  Tv 

1Q          17 

16.  14^. 

11.  ^r- 

1/1        434 

17     O     3 

Lt*     ^1000* 

12.  t&t- 

15.  tWoV- 

1Q      1     6444 

10*  J-ioooou- 

19.  Ninety-eight  hundredths. 

20.  One  hundred  twenty-five  thousandths. 

21.  One  thousand  eighteen  ten-thousandths. 

22.  Six,  and  seventy-five  hundredths. 

23.  Two  thousand  four  hundred  sixty-two,  and  seven 
tenths. 

24.  Four  hundred,  and  forty-four  thousandths. 

25.  Twenty-four,  and  twenty-four  millionths. 

26.  Thirty-five  thousand  three  hundred  fifty-one,  and 
twenty-one  hundredths. 

27.  One  million  two  hundred  thousand  thirty,  and 
one  millionth. 


178  DECIMALS. 

SECTION    XXXVII. 

ADDITION  AND  SUBTRACTION  OF 
DECIMALS. 

382.— 1.  What  is  the  sum  of  3  tenths  and  5  tenths? 

2.  What  is  the  sum  of  7  tenths  and  8  tenths  ? 

Solution. — 7  tenths  and  8  tenths  are  15  tenths ;  and  since  10 
tenths  are  equal  to  1,  15  tenths  are  1  and  5  tenths,  or  1.5. 
Hence,  the  sum  of  7  tenths  and  8  tenths  is  1.5. 

3.  What  is  the  sum  of  12  hundredths  and  11  hun- 
dredths? 

4.  If  you  pay  25  hundredths  of  a  dollar  for  a  slate, 
and  10  hundredths  of  a  dollar  for  some  paper,  what  part 
of  a  dollar  do  you  pay  for  the  whole  ? 

5.  If  you  pay  25  hundredths  of  a  dollar  for  a  slate, 
and  10  hundredths  of  a  dollar  for  some  paper,  how 
much  more  does  the  slate  cost  than  the  paper  ? 

6.  If  you  have  75  hundredths  of  a  bushel  of  chest- 
nuts, and  your  brother  has  15  hundredths  of  a  bushel, 
how  many  more  have  you  than  your  brother  ? 

383.  Principle. — Decimals  which  express  like  parts  of 
a  unit  may  be  added  or  subtracted  like  integers. 

WRITTEN  EXERCISES. 

384.— 1.  Add  13.634,  35.423  and  8.56. 

hi.b34  Solution. — Write  the  numbers  so  that  figures 

fjo.Jf.2S  of  the  same  order  shall  stand  in  the  same  column. 

8.50  Add  as  in  addition  of  integers,  which  gives  57.617, 

57 6 17  ^e  resu^  required. 

2.  Add  .315,  17.563  and  63.05. 


DECIMALS.  179 

3.  From  963.75  subtract  585.125. 

963,750  Solution. — Write  the  numbers  so  that  figures 

ron  -igc       of  the  same  order  shall  stand  in  the  same  column. 

- Subtract  as  in  subtraction  of  integers,  which 

378.625       gives  378,625,  the  result  required. 

These  numbers  are  prepared  for  subtracting  by  annexing  a 
cipher  to  the  minuend,  which  makes  its  decimal  express  thou- 
sandths, or  like  parts  with  the  decimal  of  the  subtrahend,  with- 
out altering  the  value. 

4.  From  196.35  subtract  173.91. 

5.  From  73.007  subtract  68.75. 

385.  Rule  for  Addition  and  Subtraction  of  D3cimals.—  Write 
the  numbers  so  that  figures  of  the  same  order  shall 
stand  in  the  same  column. 

Add  or  subtract  in  the  same  manner  as  in  in- 
tegers,  and  place  the  decimal  point  at  the  left  of 
the  order  of  tenths  in  the  result. 

PROBLEMS. 

386.— 1.  What  is  the  sum  of  .81,  3.65,  4.5  and  7.315? 

2.  What  is  the  sum  of  .9,  14.501,  6.75  and  19? 

3.  What  is  the  sum  of  .125  +  .62  +.437  ? 

4.  What  is  the  difference  between  41.634  and  7.595? 

5.  What  is  the  difference  between  18.5  and  9.995? 

6.  What  is  the  difference  between  .735  of  a  ton  and 
.598  of  a  ton? 

7.  What  is  the  difference  between  98  and  .98  ? 

8.  Wilson  is  63.125  years  old,  and  Johnson  is  58.75 
years  old.     What  is  the  sum  of  their  ages  ? 

9.  Horatio  has  125.675  thousand  feet  of  box-boards, 
and  his  father  239.703  thousand  feet.  How  many  more 
has  the  one  than  the  other? 


180  DECIMALS. 

10.  A  farmer  received  $69,875  for  corn,  $93.1875  for 
wheat,  and  $42.9375  for  oats.  How  much  did  he  re- 
ceive for  the  whole  ? 


387.  Test  Questions. — 1.  What  is  a  decimal?     A  mixed 
decimal  ? 

2.  Kecite  the  decimal  orders  from  tenths  to  millionths.     What 
principles  of  decimals  are  given? 

3.  How  do   you   read   decimals?     How  do   you   write  deci- 
mals ? 

4.  What  is  the  principle  for  addition  and  subtraction  of  deci- 
mals ?     How  do  you  add  or  subtract  decimals  ? 


SECTION   XXXVIII. 
MULTIPLICATION  OF  DECIMALS. 

388. — 1.  How  many  tenths  are  3  times  1  tenth?  3 
times  3  tenths  ?     5  times  4  tenths  ? 

2.  How  many  ones  are  5  times  4  tenths  ?  5  times  9 
tenths  ? 

Solution. — 5  times  9  tenths  are  45  tenths;  and  since  10 
tenths  are  1,  45  tenths  are  4  ones  and  5  tenths,  or  4.5. 

3.  How  many  hundredths  are  3  times  1  hundredth  ? 
3  times  3  hundredths  ?     5  times  4  hundredths  ? 

4.  How  many  tenths  are  5  times  4  hundredths  ? 

5.  When  tenths  are  multiplied  by  ones,  what  is  the 
denominator  of  the  product?  When  hundredths  are 
multiplied  by  ones,  what  is  the  denominator  ? 

6.  What  is  the  product  of  1  tenth  by  1  tenth,  or  of 
rV  hy  ^  ?     Of  3  tenths  by  2  tenths,  or  T\  by  ^  ? 


DECIMALS.  181 

7.  What  is  the  product  of  1  hundredth  by  1  tenth,  or 
ru  (T  by  tV  ?     Of  3  hundredths  by  2  tenths,  or  -j-fo  by  ^  ? 

8.  When  tenths  are  multiplied  by  tenths,  what  is  the 
denominator  of  the  product?  When  hundredths  are 
multiplied  by  tenths  ? 

9.  How  many  decimal  orders  will  there  be  in  the 
product,  decimally  expressed,  if  you  multiply  rcr  by  3  ? 
iVbyT3<r?    rbby3?    ykby^? 

389.  Principle. — The  number  of  decimal  orders  in  the 
product  is  equal  to  the  number  of  decimal  orders  in  both 
factors. 

WRITTEN  EXERCISES. 

330.— 1.  What  is  the  product  of  .49  by  6  ? 

.49 

6         Solution.— 6  times  T4o9o  are  ^^  =  flft  ===  2T%%  or 

decimally  expressed,  2.94. 

2.  What  is  the  product  of  .49  by  .6? 

.49 

6         Solution.— A  times  T%%  are  T%\  X  &  =  ToUfo  = 
tVA;  or,  decimally  expressed,  .294. 


.294 


3.  What  is  the  product  of  .573  by  5  ? 

4.  What  is  the  product  of  .75  by  .06  ? 

391.  Rule  for  Multiplication  of  Decimals.— Multiply  as  in 
integers,  and  point  off  as  many  orders  for  decimals 
in  the  product  as  there  are  orders  of  decimals  in 
both  factors. 

If  there  are  not  as  many  figures  in  the  product  as 
decimal  orders  required,  supply  the  deficiency  by 
prefixing  ciphers. 

16 


182 


DECIMALS. 


PR  OBL  EMS. 

392.  Multiply— 

5.  4.3  by  .7 


1.  35  by  .5 

2.  35  by  .05 

3.  3.5  by  5. 

4.  .35  by  5. 


9.  4.06  by  1.2 

10.  4.06  by  .12 

11.  .406  by  12. 

12.  40.6  by  1.2 


6.  .43  by  .07 

7.  4.3  by  7. 

8.  43  by  .7 

13.  The  distance  to  a  certain  place  is  .75  of  a  mile; 
to  another  place  it  is  10  times  as  far.  What  is  the  dis- 
tance to  the  latter  place  ? 

Since  the  multiplier  is  10,  the  product  may  be  found  at  once 
by  removing  the  decimal  point  in  the  multiplicand  one  order 
to  the  right. 

14.  The  standard  bushel  contains  2150.42  cubic 
inches.  What  is  the  cubic  capacity  of  a  bin  containing 
100  bushels? 

15.  What  is  the  product  of  .00365  by  1000? 

16.  What  will  10.25  yards  of  cloth  cost,  at  $3.40  per 
yard? 


SECTION    XXXIX. 
DIVISION   OF  DECIMALS. 

393. — 1.  How  many  times  3  tenths  in  9  tenths? 

2.  How  many  times  3  hundredths  in  9  tenths? 

Solution. — 9  tenths  are  90  hundredths;  3  hundredths  are 
contained  in  90  hundredths  30  times. 

3.  How  many  times  2  hundredths  in  8  tenths  ?     In 
6  tenths  ? 

4.  What  is  the  quotient  of  9  tenths  by  3  tenths,  or  f\j 

bytV?    Of  A  by  A? 


DECIMALS.  183 

5.  What  is  the  quotient  of  8  tenths  by  2  hundredths, 
or^by^?     Of^byy^? 

6.  What  is  the  quotient  of  5  tenths  by  25  hundredths, 
or  .5  by  .25?     Of  1.2  by  .12? 

7.  What  is  the  quotient  of  4.2  by  6,  or  what  is  one 
sixth  of  4.2  ? 

Solution. — 4.2  is  42  tenths ;  and  one  sixth  of  42  tenths  is  7 
tenths. 

8.  What  is  the  quotient  of  .42  by  6  ?  Of  .42  by  .06  ? 
Of  6  by  .6? 

9.  When  .42  is  divided  by  6,  how  many  more  orders 
of  decimals  has  the  dividend  than  the  divisor  ? 

10.  When  .42  is  divided  by  6,  how  many  orders  of 
decimals  has  the  quotient  ? 

394.  Principle. — The  number  of  decimal  orders  in  the 
quotient  is  as  many  as  there  are  in  the  dividend  less  the 
number  in  the  divisor. 

WKITTEN   EXERCISES. 

395.— 1.  How  many  times  6  in  2.94  ? 

6)2.94         Solution. — 2.94  is  294  hundredths,  and  one  sixth 
^g      of  294  hundredths  is  49  hundredths,  or  .49. 

2.  How  many  times  .6  in  .294? 

Solution. — .6  is  the  same  as  TV  of  6 ;  6  is  con= 
£)  .294      tained  in  .294  one  sixth  of  .294,  or  .049,  times ;  and 
.49      A  of  6  must  be  contained  10  times  .049  times,  or 
.49  times. 


3.  How  many  times  9  in  28.35? 
4  How  many  times  .07  in  .0938  ? 


184 


DECIMALS. 


396.  Rule  for  Division  of  Decimals.— Divide  as  in  inte- 
gers, and  point  off  in  the  quotient  as  many  deci- 
mal orders  as  those  in  the  dividend  exceed  in 
number  those  in  the  divisor. 

If  there  are  not  as  many  figures  in  the  quotient 
as  the  number  of  decimal  orders  required,  sup-ply 
the  deficiency  by  prefixing  ciphers. 


397.  Divide— 

PROBL  EMS. 

1.  6.15  by  5. 

5.  .0075  by  .15 

9. 

32.76  by  78 

2.  30.1  by  .7 

6.  99  by  .99 

10. 

60  by  .8 

3.  .825  by  .25 

7.  .99  by  99. 

11. 

99  by  9.9 

4.  4.06  by  1.2 

8.  9.9  by  .99 

12. 

.0738  by  .6 

13.  What  is  the  quotient  of  75.6  divided  by  100? 

Since  the  divisor  is  100,  the  quotient  is  obtained  at  once  by 
removing  the  decimal  point  in  the  dividend  two  orders  to  the 
left. 

14.  What  is  the  quotient  of  31.45  divided  by  1000? 

15.  What  is  the  quotient  of  .905  divided  by  10000? 

16.  What  is  the  quotient  of  23  by  .3  to  2  orders  of 

decimals  ? 

Solution. — Annex  ciphers  as  decimal  orders, 

.3)23.000        and  continue  the  division  till  a  quotient  is  ob- 

76.66 -t-    tained  with  the  required  number  of  decimal 

orders,  which  is  76.66  +. 

The  sign  -J-  is  annexed  to  the  result  to  indicate  that  the 

division  is  incomplete,  and  could  have  been  carried  farther. 

17.  What  is  the  quotient  of  .025  by  .41  to  4  orders 
of  decimals? 

18.  How  many  yards  of  cloth  at  $.17  a  yard  can  be 
purchased  for  $2,635  ? 


in  A? 


DECIMALS.  185 

SECTION    XL. 
REDUCTION  OF  DECIMALS. 

CASE   I. 

Decimals  Reduced  to  Common  Fractions. 
398, — i.  What  does  .5  express?     How  many  halves 

TIP 

2.  What  does  .6  express?     How  many  fifths  in  -&? 

3.  What  does  .75  express?    How  many  fourths  in  yW? 

4.  What  does  .45  express?     How  many  twentieths 

in       4  5    9 

ln  TOT  • 

5.  What  does  .045  express  ?     How  many  hundredths 

WRITTEN  EXERCISES. 

399. — 1.  Reduce  .25  to  a  common  fraction. 

ge--  &  j.  *  -_-  *         Solution. — .25  may  be  expressed  in 
100      20      4       the  form  of  a  common  fraction  by  omit- 
ting the  decimal  point  and  writing  the  denominator,   which 
gives  T%\. 

■f5\  reduced  to  its  lowest  terms  is  J,  which  is  the  fraction  re- 
quired. 

2.  Reduce  .05  to  a  common  fraction. 

3.  Reduce  .056  to  a  common  fraction. 

4.  Reduce  .125  to  a  common  fraction. 

5.  Reduce  .625  to  a  common  fraction. 

400.  Rule  for  Reduction  of  Decimals  to  Common  Fractions.— 

Omit  the  decimal  point;  write  the  denominator 
under  the  given  numerator,  and  reduce  the  frac- 
tion to  its  lowest  terms. 

16* 


186  DECIMALS. 

I* R  OB  L  EMS. 

401.  Reduce  to  common  fractions  in  lowest  terms— 


1. 

.85 

4.  .016 

7.  .096 

2. 

.125 

5.  .625 

8.  .008 

3. 

.025 

6.  .0125 

9.  1.275 

CASE  II. 

Common  Fractions  Reduced  to  Decimals. 

402—  1.  How  many  tenths  in  i?     In  f  ?     In  £ 

2.  How  many  hundredths  in  \  ?     In  \  ?     In  f  ? 

3.  How  many  hundredths  in  |  ?     In  ^  ?     In  ^ 

WRITTEN  EXERCISES. 

403. — 1.  Reduce  i  to  a  decimal. 


8" 

8)3.000         Solution.— f  is  \  of  3. 

— ^ZZ  3  is  30  tenths,  or  3.0 ;  and  J  of  30  tenths  is  3 

tenths,  or  .3,  with  6  tenths  remaining. 

6  tenths  are  60  hundredths ;  and  J  of  60  hundredths  is  7  hun- 
dredths, or  .07,  with  4  hundredths  remaining. 

4  hundredths  are  40  thousandths ;  and  |  of  40  thousandths  is 
5  thousandths,  or  .005. 

The  result  is  3  tenths  7  hundredths  5  thousandths,  or  .375, 
which  is  the  decimal  required. 

2.  Reduce  f  to  a  decimal. 

3.  Reduce  -fa  to  a  decimal. 

404.  Rule  for  Reduction  of  Common  Fractions  to  Decimals.— 

Reduce  the  numerator  to  tenths,  hundredths,  etc., 
by  annexing  ciphers;  divide  this  result  by  the 
denominator,  and  point  off  as  many  orders  for 
decimals  in  the  quotient  as  there  are  ciphers 
annexed. 


DECIMALS. 


187 


PROBLEMS. 

405.  Reduce  to  decimals — 

i.  a- 

4.  yfr. 

7. 

12 
T2T- 

2.  |. 

5.  i 

8. 

fir- 

3.  &. 

6.  rV. 

9. 

Itt. 

10.  Reduce  ^  to  a  decimal  of  3  orders. 
11)^  000  Solution. — Reduce  the  3  to  thousandths 


by  annexing  three  ciphers  and  divide  by  11. 
Denote  that  the  decimal,  with  the  required 


.272  + 
orders,  is  not  exact  by  annexing  the  sign  -f. 

11.  Reduce  T5^  to  a  decimal  of  4  orders 


406.  Test  Questions. — 1.  What  is  a  principle  of  multipli- 
cation of  decimals  ?     How  do  you  multiply  in  decimals  ? 

2.  What  is  a  principle  of  division  of  decimals?     How  do  you 
divide  in  decimals? 

3.  How  do  you  reduce  decimals  to  common  fractions?    Com- 
mon fractions  to  decimals? 


SECTION   XLI. 
REVIEW  OF  DECIMALS. 

WRITTEN  EXERCISES. 

407. — 1.  Express  .0375  as  a  common  fraction. 

2.  Express  tthjW  as  a  decimal. 

3.  What  part  of  a  dollar  is  $.125  ? 

4.  What  is  the  value  of  -£$,  expressed  as  a  decimal  of 
3  orders  ? 

5.  A  farmer  sold  15  cows  for  $828.75  ;  how  much  did 
he  get  for  each  ? 


188  PERCENTAGE. 

6.  On  one  car  there  are  15.365  thousand  feet  of 
boards,  and  on  another  16.491  thousand  feet.  How 
many  thousand  feet  are  there  on  both  cars  ? 

7.  By  the  census  of  1870,  there  were  in  Rhode  Island, 
206  persons  to  a  square  mile;  in  Massachusetts,  185.6 
persons  to  a  square  mile.  How  many  more  persons  to 
a  square  mile  were  there  in  Rhode  Island  than  in  Mas- 
sachusetts ? 

8.  What  is  the  value  of  9765  pounds  of  fish,  at 
$5.20  per  hundred  ? 

9.  What  must  be  paid  for  5305  feet  of  boards,  at 
$11.40  per  thousand  feet? 

10.  How  many  feet  of  boards  can  be  bought  for 
$60,477,  at  $11.40  per  thousand  feet? 

11.  How  much  is  .005  divided  by  .00005? 

12.  A  cubic  foot  of  white  oak  weighs  42.937  pounds ; 
what  is  the  weight  of  50  cubic  feet  ? 


SECTION  XLII. 
NOTATION  OF  PERCENTAGE. 

408. — 1.  How  much  is  y^-  of  100  pounds?     y-f-^  of 
100  pounds  ?     jib  of  100  pounds  ? 

2.  How  much  is   y^  of  400  yards?     yfo  of  400 
yards  ?     T^  of  400  yards  ? 

3.  What  is  a  per  cent,  of  a  number  ?     A  per  cent,  of 
a  number  is  a  number  of  hundredths  of  that  number. 

4.  How  many  per  cent,  of  a  number  is  T-^-0-?     Is 

3     9 


PER  CENT  A  GE.  189 

5.  To  take  4  per  cent,  of  a  number,  is  to  take  how 
many  hundredths  of  the  number? 

6.  What  part  of  a  number  is  2  per  cent,  of  a  number? 

Solution. — 2  per  cent,  is  Tf -q,  or  -^ ;  hence,  2  per  cent,  of  a 
number  is  -^  of  the  number. 

7.  What  part  of  a  number  is  4  per  cent,  of  it?     5 
per  cent,  of  it  ?     10  per  cent,  of  it  ? 

8.  What  part  of  a  number  is  20  per  cent,  of  it?     40 
per  cent,  of  it  ?     50  per  cent,  of  it  ? 

9.  What  per  cent,  of  a  number  is  expressed  by  T\°o  ? 

ByiV*?     By  mi 

10.  What  per  cent,  of  a  number  is  .06  of  it?     Is  .30 
of  it?     Is  .75  of  it?     Is  1.25  of  it? 

DEFINITIONS. 

409.  Any  Per  Cent,  of  a  number  is  so  many  hun- 
dredths of  the  number. 

The  term  per  cent,  is  a  contraction  of  per  centum,  which 
means  by  the  hundred. 

410.  The  Rate  per  cent,  is  the  number  denoting  the 
number  of  hundredths. 

411.  The  Sign  of  per  cent,  is  %,  and  is  read  per  cent. 
Thus, 

j^>  or  .01,  may  be  written  1%,  and  read  one  percent. 

lobyOT-06>        "        "        6#,      "  six  per  cent 

4         i 

~J^~0  y  or  .  04 2  '      "        "        4  J  % ,     "    four  and  one  half  per  cent 

115  1      1C 

■jm  >  or  1 . 1 5f      "        •<        H5  fc  9   «  one  hundred  fifteen  per  cent. 


190 


PERCENTAGE. 


4:12.  The  Base  is  the  number  or  quantity  of  which  the 
per  cent,  is  computed. 

413.  Percentage  is  the  process  of  computing  by  hun- 
dredths. 

The  term  Percentage  is  applied,  also,  to  that  part  of 
the  base  which  corresponds  to  the  rate  per  cent. 


WRITTEN   EXERCISES. 


414:.  Express  decimally- 


1. 

3%. 

5. 

4f%. 

9. 

45%. 

2. 

4%. 

6. 

Hfb. 

10. 

67%. 

3. 

7%. 

7. 

17*%. 

11. 

125%. 

4. 

9%. 

8. 

7i%- 

12. 

135%. 

SECTION   XLIII. 
CASES  IN  PERCENTAGE. 


CA.SE   I. 

Base  and  Rate  Per  Cent,  given,  to  find  the  Percentage. 

415. — 1.  What  part  of  a  number  is  4%  of  it? 

Solution. — 4%  is  T$7,  or  ^;  hence,  4%  of  a  number  is  j% 
of  the  number. 

2.  What  part  of  a  number  is  15%  of  it?    30%  of  it? 

3.  What  part  of  a  number  is  20%  of  it?     12%  of 
it?     50%  of  it? 

4.  What  is  the  simplest  form  of  20%  ? 

Solution.— 20%  is  &%,  which  in  its  simplest  form  is  J. 


PERCENTAGE.  191 

5.  What  is  the  simplest  form  of  50%  ?     Of  60%  ? 
Of  80%? 

6.  What  is  the  simplest  form  of  12£%  ?     Of  16f  %  ? 
Of  33^%  ? 

7.  A  farmer  had  20  cows,  and  sold  10%  of  them. 
How  many  did  he  sell  ? 

8.  If  from  a  flock  of  sheep  5%  of  them  should  be 
taken,  what  per  cent,  would  remain  ? 

9.  A  man  bought  a  horse  for  $100,  and  sold  him  at  a 
profit  of  25%.     For  what  sum  did  he  sell  him? 

10.  If  you  buy  a  horse  for  $100,  and  sell  him  at  a 
loss  of  20%,  how  much  do  you  get  for  him? 

WRITTEN  EXERCISES. 

416.— 1.  What  is  7%  of  9546  yards? 

9546  yds.  Solution.— Since  7%   is  .07,  7%  of  9546 

.07  yards  is  .07  of  9546  yards,  which  is  668.22  yards, 


668.22  yds.       the  Percentage  required. 

2.  What  is  5%  of  860  pounds? 

3.  What  is  12%  of  750  men? 

417.  Rule  for  Finding  any  Per  Cent,  of  a  Number.— Multi- 
vly  the  given  number  by  the  rate  per  cent,  expressed 
decimally. 

PROBLEMS. 

418.  How  much  is — 

1.  4%  of  $61.50?  7.  19%  of  562? 

2.  3|%  of  $974?  8.  12%  of  880? 

3.  7±%  of  160yd.?  9.  10%  of  $750? 

4.  25%  of  530rd.?  10.  8%  of  450  men? 

11.  15%  of  560cwt.? 

12.  60%  of  7251b.? 


5.  lf%  of  9160? 

6.  3£%  of  850? 


192  PERCENTAGE. 

13.  If  you  earn  in  a  year  $300,  and  spend  45%  of 
that  sum,  how  much  do  you  spend  ? 

14.  What  is  a  merchant's  commission  for  selling 
goods  to  the  amount  of  $7500,  at  2£%  ? 

15.  Of  a  cargo  of  corn,  consisting  of  2560  bushels, 
15%  was  damaged.     How  much  was  not  damaged? 

16.  If  a  town  having  9540  inhabitants  should  in- 
crease in  population  20%,  what  would  be  the  number 
of  its  inhabitants  ? 

CASE   II. 

Base  and  Percentage  given,  to  find  the  Rate  Per  Cent. 

419. — 1.  What  per  cent,  of  a  number  is  |  of  it? 

2.  What  part  of  5  is  2  ?     What  per  cent,  of  1  is  f  ? 

3.  I  spent  \  of  my  money.  What  per  cent,  of  it  did 
I  spend  ? 

4.  I  had  $20,  and  spent  $5.  What  per  cent,  of  my 
money  did  I  spend  ? 

Solution. — I  had  $20,  and  spent  $5;  hence,  I  must  have 
spent  s5o,  or  i  of  my  money;  and  \  equals  T2o5o,  or  25%. 

5.  What  per  cent,  of  $10  is  $2  ?     Of  $45  is  30  ? 

6.  8  days  are  what  per  cent,  of  20  days  ?     Of  48  days  ? 

7.  I  received  $3  for  collecting  $60.  Wnat  per  cent, 
did  I  receive  ? 

8.  A  farmer  bought  a  cart  for  $50,  and  sold  it  for 
$60.     What  per  cent,  did  he  gain  ? 

9.  James  bought  a  knife  for  $.60,  and  sold  it  for  $.48. 
What  per  cent,  was  the  loss  ? 

10.  What  per  cent,  does  a  merchant  gain  who  buys 
flour  at  $8  per  barrel,  and  sells  it  at  the  rate  of  3  barrels 
for  $30? 


PERCENTAGE.  198 

WRITTEN    EXERCISES. 

420.— -1.  What  per  cent,  of  400  is  24  ? 

400)2 A.00(. 06  =  6°Io         Solution.— Since  24  is  jfo  of 
2 1  go  400,  it  is  .06,  or  6?0,  of  400,  which 

is  the  rate  per  cent,  required. 

2.  What  per  cent,  of  520  is  26  ? 

3,  What  per  cent,  of  95  bushels  are  3.6  bushels  ? 

421.  Rule  for  finding  the  Rate  Per  Cent,  that  a  given  Percentage 
is  of  the  Base— Divide  the  percentage  by  the  base,  ex- 
tending the  quotient  to  hundredths. 

PROBLEMS. 

422.  What  per  cent,  of— 

1.  54  yards  are  9  yards? 

2.  125  men  are  8  men? 

3.  $650  are  $39.25  ? 

4.  530rd.  are  132ird.  ? 

5.  9160  is  160.3? 

6.  400  is  28  ? 


7.  560  is  106.4? 

8.  725  is  435  ? 

9.  $756  are  $75  ? 

10.  450  men  are  36  men  ? 

11.  560cwt.  are84cwt.? 

12.  7251b.  are  4351b.  ? 

13.  If  you  earn  $300,  and  spend  $135  of  it,  what 
rate  per  cent,  of  the  sum  earned  is  the  sum  spent  ? 

14.  A  house,  which  cost  $5670,  was  sold  for  $283.50 
above  the  cost.     What  was  the  rate  per  cent,  of  gain  ? 

15.  Of  an  army  of  45450  men,  12626  men  were  lost 
in  battle.     What  was  the  rate  per  cent,  of  loss  ? 

16.  If  a  horse,  which  cost  $250,  was  sold  for  $100 
above  cost,  what  was  the  rate  per  cent,  gained  ? 

17.  A  certain  town,  which  had  9540  inhabitants,  has 
increased  by  1908  persons.  What  is  the  rate  per  cent, 
of  increase? 

17 


194 


PERCENTAGE. 


SECTION   LIV. 


INTEREST. 

423. — 1.  When  the  allowance  for  the  use  of  money 
is  6%,  how  many  cents  is  it  for  $1,  or  100  cents?  How 
much  is  it  for  $100  ? 

2.  When  the  allowance  for  the  use  of  $1  is  6  cents, 
what  is  the  rate  per  cent.  ?     When  it  is  5  cents  ? 

3.  When  the  allowance  for  the  use  of  $100  for  1 
year  is  $6,  what  is  the  yearly  rate  per  cent.  ?  When  it 
is  $5,  what  is  the  yearly  rate  per  cent.  ? 

4.  When  the  yearly  allowance  for  the  use  of  money 
is  at  the  rate  of  6%,  what  is  the  allowance  for  the  use 
of  $5  for  3  years  ? 

Solution.— The  allowance  for  1  year  at  6%  is  t|q  of  the 
base;  hence,  the  allowance  for  the  use  of  $5  for  ]  year  must  be 
t^;0  of  $5,  or  $.30,  and  for  the  use  of  $5  for  3  years  must  be  3 
times  $.30,  or  $.90. 


PERCENTAGE.  195 

5.  What  is  the  allowance  for  the  use  of  $8  for  4 
years,  at  a  yearly  rate  of  7  %  ? 

6.  What  is  the  allowance  for  the  use  of  $25  for  5 
years,  at  a  yearly  rate  of  4%  ? 

7.  What  is  the  allowance  for  the  use  of  $50  for  4 
months,  at  a  yearly  rate  of  6  %  ? 

Solution. — The  allowance  for  the  use  of  $50  for  1  year  at  6 
per  cent,  is  $3 ;  hence,  for  4  months,  or  £  of  a  year,  it  must  be  J 

of  $3,  or  $1. 

8.  What  is  the  allowance  for  the  use  of  $40  for  3 
months,  at  a  yearly  rate  of  7  %  ? 

9.  What  is  the  allowance  for  the  use  of  $90  for  2 
months,  at  a  yearly  rate  of  6%  ? 

10.  In  computing  the  allowance  for  the  use  of  money, 
a  month  is  commonly  regarded  as  how  many  days  ? 

Ans.  30  days. 

11.  What  is  the  allowance  for  the  use  of  $200  for  1 
month  20  days,  at  a  yearly  rate  of  6  %  ? 

Solution. — The  allowance  for  the  use  of  $200  for  1-  year  is 
$12 ;  hence,  for  1  month,  or  TV  of  a  year,  it  is  TV  of  $12,  or  $1, 
and  for  20  days,  or  f  of  a  month,  it  is  f  of  $1,  or  $.66f ;  $1  and 
$.66f  is  $1.66f,  the  allowance  required. 

12.  What  is  the  allowance  for  the  use  of  $100  for  2 
months  15  days,  at  a  yearly  rate  of  8%  ? 

13.  What  is  the  allowance  for  the  use  of  $300  for  1 
month  10  days,  at  a  yearly  rate  of  5%  ? 

14.  What  is  the  allowance  for  the  use  of  $120  for  1 
year  4  months  6  days,  at  a  yearly  rate  of  5%  ? 

15.  What  is  the  allowance  for  the  use  of  $200  for  2 
years  5  months  15  days,  at  a  yearly  rate  of  12%  ?  At  a 
yearly  rate  of  16|%  ? 


196  PERCENTAGE. 

DEFINITIONS. 

424.  Interest  is  an  allowance  for  the  use  of  money  or 
its  equivalent. 

425.  The  Principal  is  the  sum  for  the  use  of  which  in- 
terest is  allowed. 

426.  The  Rate  of  Interest  is  the  rate  per  cent,  of  the 
principal  allowed  for  its  use  for  one  year. 

427.  The  Amonnt  is  the  sum  of  the  principal  and  in- 
terest. 

WRITTEN  EXERCISES. 

428.— 1.  What  is  the  interest  of  $345.50  for  3  years 
3  months,  at  6%  ? 

Principal,                       $345.50  SoLUTiON.-The  inter- 

d   .                                                 qq  est  of  $345.50  for  1   year 

„ is    .06    of   that    sum,   or 

Int.  for  ly.,                 §20.7300  $20.73  .  hence>  for  3  years 

&_  the    interest    is    3    times 

Int.  for  3y.,                   $62.19  #20.73,  or  $62.19. 

,     \      /              ,              _,  iol  The  interest  of  $345.50 

Int.  for  Smo.,  or  jy.9        5.18-  foY  1  year  .g  $2Q  n .  hence> 

Int.  for  3y.  3mo.y       $67. 37  j        for  3  months>  or  *  of  a 
J         *  *         year,  it  is  \  of  $20.73,  or 

$5.18J.     The  interest  for  3  years  3  months  is  $62.19  +$5.18|,  or 

$67.37J. 

2.  What  is  the  interest  of  $500  for  5  years  6  months,, 
at  7%? 

3.  What  is  the  interest  of  $750  for  4  years  5  months^ 
at  8%? 

4.  What  is  the  interest  of  $840  for  3  years  4  months, 
at  6%? 

5.  What  is  the  interest  of  $920  for  4  years  2  months, 
at  5%  ? 


PERCENTAGE. 


197 


6.  What  is  the  amount  of  $450  for  7  months  20  days, 

at  4%? 

Solution. — The  inter- 
est of  $450  for  1  year  is 
.04  of  the  principal,  or 
-°4  $18 ;  hence,  for  6mo.,  or  } 
year,  it  is  i  of  $18,  or  $9; 
for  lmo.,  or  \  of  6mo.,  it 
is  \  of  $9,  or  $1.50;  and 
for  20d.,  or  fmo.,  it  is  f 
of  $1.50,  or  $1;  hence, 
for  7mo.  20d.  it  is  $9  + 
$1.50  +  $1,  or  $11.50. 

The  amount  is  the  sum 
of  the  principal  and  in- 
terest, $450  +  $11.50, 
which  is  $461.50. 


$18.00 
$9.00 


Principal, 

Hate, 

Int.  for  ly., 

Int.  for  Gmo.y  or  ~y., 

Int.  for  lmo.,  or  j  of  6mo.,     1.50 

Int.  for  gOd.,  or  fmo.s  1.00 

Int.  for  7 mo.  20d.,  $11.50 

Principal, 

Amount, 


450 
$461.50 


7.  What  is  the  amount  of  $600  for  5  months  12  days, 
at7%? 

8.  What  is  the  amount  of  $720  for  2  months  3  days, 
at6%? 

9.  What  is  the  interest  of  $5000  for  25  days,  at  8%  ? 
10.  What  is  the  interest  of  $4200  for  18  days,  at  6%  ? 


429.  Rules  for  Computing  Interest.— Multiply  the  prin- 
cipal by  the  rate  of  interest  expressed  decimally, 
and  that  product  by  the  time  expressed  in  years. 

If  there  be  months  or  days,  find  the  interest  for 
the  number  of  months,  by  taking  fractional  parts 
of  a  year's  interest;  and  for  the  number  of  days, 
by  talcing  fractional  parts  of  a  month's  interest. 

To  find  the  amount,  add  the  principal  and  in- 

terest. 
17* 


198  PERCENTAGE. 

PMOBLEMS. 

430. — What  is  the  interest  of — 

1.  $1800  for  3  years  9  months,  at  7%  ? 

2.  $34.50  for  1  year  6  months,  at  6%  ? 

3.  $75.50  for  6  years  11  months,  at  8%? 

4.  $5600  for  4  years  5  months,  at  5%  ? 

5.  $8000  for  2  years  2  months,  at  4%  ? 

6.  $10000  for  15  years  1  month,  at  3|%? 

7.  $240  for  10  months  6  days,  at  9%? 

8.  $6450  for  3  months  3  days,  at  10%  ? 

9.  $65.40  for  1  month  24  days,  at  6%? 

10.  What  is  the  amount  of  $725  for  1  year  6  months 
10  days,  at  8%? 

11.  What  is  the  amount  of  $830  for  2  years  3  months 
27  days,  at  7%  ? 

12.  What  is  the  amount  of  $300  for  18  days,  at  6%  ! 

13.  What  is  the  interest  of  $407.60  from  Jan.   1, 

1869,  to  April  1,  1871,  at  6%? 

14.  What  is  the  amount  of  $1000  from  May   15, 

1870,  to  October  18,  1871,  at  7%? 

15.  What  is  the  amount  of  $1200  from  February  1, 

1871,  to  November  1,  1872,  at  8%  ? 


431.  Test  Questions. — 1.  What  is  any  per  cent,  of  a  num 
ber?  What  is  the  rate  per  cent.?  The  base?  What  is  per- 
centage ? 

2.  How  do  you  find  any  per  cent,  of  a  number?  The  rate  per 
cent,  a  given  percentage  is  of  a  base? 

3.  What  is  interest?  The  principal?  The  rate  of  interest? 
The  amount?     How  do  you  compute  interest? 


REVIEW.  199 

SECTION    XLV. 
REVIEW  OF  PERCENTAGE. 

432. — 1.  What  is  a  broker's  charge,  at  \%,  for  buy- 
ing stocks  amounting  to  $400  ? 

2.  An  agent  sells  goods  to  the  amount  of  $500,  and 
reserves  2^%  of  the  amount  for  his  services.  How 
much  did  he  receive  ? 

3.  I  bought  broadcloth  for  $5  per  yard,  and  sold  it 
at  20  %  above  cost.     How  much  did  I  gain  per  yard  ? 

4.  I  bought  cloth  at  $5  per  yard,  and  sold  it  so  as  to 
gain  $1  per  yard.     What  rate  per  cent,  was  the  gain  ? 

5.  I  bought  a  watch  for  $60,  and  sold  it  at  a  loss  of 
10%.     How  much  did  I  get  for  it? 

6.  At  what  price  must  I  sell  a  watch  that  cost  me  $50 
to  gain  20%  ? 

7.  If  I  sell  a  watch  which  cost  me  $60,  at  a  loss  of 
20%,  how  much  do  I  get  for  it? 

8.  How  many  dollars  are  $20  less  5%  of  $20? 

9.  I  sold  a  carriage  which  cost  me  $80  for  $60. 
What  was  the  rate  per  cent,  of  loss  ? 

10.  If  you  sell  what  cost  you  $6  for  $8,  what  is  the 
rate  per  cent,  of  gain  ? 

11.  What  is  the  interest  of  $200  for  3  years  6  months, 
at  6%? 

12.  What  is  the  interest  of  $400  for  15  days,  at  6%? 

13.  What  is  the  amount  of  $300  for  4  months  at  7%  ? 

14.  What  is  the  amount  of  $50  for  6  months,  at  8%  ? 

15.  What  is  the  interest  of  $600  for  5  years  9  months, 
at  6%? 


•200  REVIEW. 

WB I TT  KX    EXER  CISE8. 

433. — 1.  How  much  must  be  paid  for  insuring  a  house 
for  $1500,  at  2^%  ? 

2.  What  is  the  interest  of  $524  for  3  months  12  days, 
at  4i%  ? 

3.  If  you  buy  wheat  at  $1.80  per  bushel,  and  sell  it 
so  as  to  gain  5  % ,  what  do  you  get  per  bushel  for  it  ? 

4.  A  school-house  which  cost  $7500  is  insured  for 
66f%  of  its  cost.     For  how  much  is  it  insured? 

5.  What  is  the  amount  of  $1004.45  for  1  year  6 
months  6  days,  at  6  %  ? 

6.  If  you  give  your  note  for  $800,  and  pay  it  2  years 
8  months  afterward,  with  interest  at  7%,  what  sum  will 
be  required  ? 

7.  If  you  should  have  a  note  for  $1200,  which  is  pay- 
able in  2  months  3  days,  without  interest,  and  in  bor- 
rowing money  upon  it  you  should  allow  interest  in 
advance,  for  the  time,  at  6%,  how  much  would  that 
interest  be  ? 

8.  I  bought  a  village  lot  for  $500,  and  sold  it  for 
$550.     What  rate  per  cent,  was  my  gain  ? 

9.  If  I  should  ask  $550  for  a  village  lot,  but  should 
fall  upon,the  price  10%,  what  would  then  be  my  price? 

10.  I  bought  a  horse  for  $250 ;  at  what  price  must  I 
sell  him  to  gain  20%? 

11.  What  sum  will  be  required  to  pay  a  note  for 
$1540,  which  has  been  on  interest  at  6%  for  3  years 
4  months  24  days  ? 

12.  I  bought  a  farm,  May  16,  1870,  for  $4670,  and 
paid  for  it,  October  22,  1871,  with  interest  at  8%. 
How  much  did  I  pay  ? 


APPENDIX. 


RECTANGULAR  MEASUREMENTS. 

434.  A  Line  is  that  which  lias  only  length. 

435.  A   Straight.  Line    is  a   line     a B 

that  has  only  one  direction. 

Tims,  the  line  A  B  is  a  straight  line. 

436.  An  Angle  is  the  difference 
of  direction  of  two  lines  that 
meet. 

Thus,  the  lines  AB  and  AC,  meeting 
at  Ay  form  the  angle  CAB. 

437.  A  Perpendicular  Line   is  a    straight    line    which 
meets  another  straight  line  so  as  to 

form  two  equal  angles. 

Thus,  (he  line  CD  is  a  perpendicular  line. 

438.  A  Right  Angle  is  an  angle 
formed  by  two  lines  perpendicular 
to  each  other. 

Tims,  the  angles  ACD  and  DCB  are  each  right  angles. 

439.  A  Rectangle  is  any  figure  which  has  four  straight 


sides  and  four  equal  angles. 

Thus,    the    figure   in   the    margin,   having   four 
straight  sides  and  four  equal  angles,  is  a  rectangle. 


r 


20  r 


202 


APPENDIX. 


\ 


440.  A    Rectangular  Solid    is 

any    volume    bounded    by   six 
rectangular  faces. 

Tims,  the    figure    in    the    margin, 


having  six  rectangular  faces,  represents  a  rectangular  solid. 


CASE  I. 

Rectangular  Surfaces. 

441. — 1.  How  many  square  feet  are  in  a  rectangular 
surface  which  is  3  feet  long  and  1  foot  broad? 

2.  How  many  square  inches  are  in  a  rectangular  strip 
of  paper  which  is  6  inches  long  and  1  inch  wide? 

3.  How  much  will  a  rectangular  board  9  feet  long 
and  1  foot  broad  cost,  at  3  cents  per  square  foot? 

DEFINITIONS. 

442.  The  Dimensions  of  a  rectangular  surface  are  its 
length  and  breadth. 

443.  The  Area  of  a  rectangle  is  the  extent  of  surface 
bounded  by  its  two  dimensions  or  sides. 

Thus,  the  rectangle  in  the  margin  will  be 
seen  to  contain  12  square  inches,  if  it  be  sup- 
posed to  be  4  inches  long  and  3  inches  wide. 

For,  upon  each  inch  of  length  (here  may  be 
conceived  to  be  1  square  inch,  making  a  row 
of  4  square  inches ;  and  as  there  will  be  as  many 
such  rows  as  there  are  inches  in  the  width, 
or  3  rows,  the  area  of  the  rectangle  must  be 
3  times  4  square  inches,  or  12  square  inches. 


WRITTEN  EXERCISES. 

444. — 1.  How  many  square  feet  of  surface  are  there 
in  a  floor  18  feet  long  and  15J  feet  wide? 


APPESDIX.  203 

18  sq.fi.  x  151  =i>7'9  *<!'/*'         Solution.— A  floor  18  feel 

long-  and  1  foot  wide  will  con- 
tain 18  square  feet  of  surface;  and  a  floor  18  feet  long  and  15^  feet 
wide  must  contain  loh  times  18  square  feet,  or  279  square  feet. 

2.  A  recta  igular  garden  14  rods  broad  contains  280 
square  rods.     How  long  is  it? 

280 +14<= 20 \  no.  of  rods  in  length.        Solution.— The  area, 

280    square   rods,   is  the 
product  of  the  length  and  breadth  of  the  garden. 

The  length,  then,  must  be  the  quotient  arising  from  the  division 
of  the  area,  280,  by  the  breadth,  14,  which  is  20  rods. 

3.  A  garden-walk  has  an  area  of  247  square  feet 
and  is  3^  feet  wide.     What  is  its  length  ? 

4.  A  rectangular  floor  is  27  feet  long  and  15  feet 
wide.  How  many  square  yards  of  carpeting  will  be 
required  to  cover  it  ? 

445.  Rules  for  Measurements  of  Rectangular  Surfaces.— 1. 
Multiply  the  length  by  the  width,  and  the  product 
will  denote  the  area. 

2.  Divide  the  area  by  either  of  the  dimensions, 
and  the  quotient  will  denote  the  other  dimension. 

PROBLEMS. 

446. — 1.  A  field  is  40  rods  long  and  35  rods  wide. 
What  is  its  area? 

2.  The  floor  of  a  square  room  measures  43  feet  on  a 
side.     What  is  its  area? 

3.  I  have  a  field  containing  3698  square  rods.  Its 
length  is  86  rods ;  what  is  its  width  ? 

4.  How  long  must  he  a  rectangular  lot  which  is  16 
rods  wide  to  contain  one  and  a  half  acres? 

5.  A  piece  of  carpeting  is  32  inches  wide  and  45 
yards  long.  How  many  square  yards  of  floor  will  it 
(•over  ? 


204 


Appendix. 


case  II. 
Rectangular  Solids. 

447. — 1.  How  many  cubic  feet  are  in  a  rectangular 
piece  of  timber  12  feet  long,  1  foot  wide,  and  1  foot 
thick? 

2.  How  many  cubic  feet  are  in  a  rectangular  beam 
20  feet  long,  1  foot  wide,  and  1  foot  thick  ? 

DEFINITIONS. 

448.  The  Dimensions  of  a  rectangular  solid  are  its 
length,  breadth,  and  thickness. 

449.  The  Cubic  Contents  of  a  rectangular  solid,  or 
volume,  is  the  extent  bounded  by 
its  feces. 


y 

/ 

s  s   /    y    ;     a 

////// 

/ 

.///// 

/ 

/ 

-a 

/ 

/ 

/ 

/ 

/^SS? 


Thus,  the  rectangular  solid,  or  volume,  in 
the  margin  will  he  seen  to  contain  72  cubic 
inches,  if  it  be  supposed  to  be  6  inches  long, 
3  inches  wide,  and  4  inches  thick. 

For,  upon  each  of  the  18  square  inches  of  the  lower  face  there  may 
be  conceived  to  be  1  cubic  inch,  making  a 
layer  of  18  cubic  inches  ;  and  as  there  will 
be  as  many  such  layers  as  there  are  inches 
of  thickness,  or  4,  the  contents  of  the 
volume  must  be  4  times  18  cubic  inches,  or  72  cubic  inches. 

WRITTEN  EXERCISES. 

450. — 1.  How  many  cubic  feet  are  there  in  a  rect- 
angular piece  of  timber  16  feet  long,  2  feet  6  inches 
broad,  and  1  foot  3  inches  thick  ? 


16  eu.fi.  x  gl  =  40  cu.  ft.         Solution.-2  ft.  6  in. 
40  cu.fl.  x  1]  =  50  eu.fi.     l  ft\3  in-  =  ^  fL 


:2£  ft.; 


APPENDIX.  205 

A  piece  of  timber  16  feet  long,  1  ft.  broad,  and  1  ft.  thick  will  con- 
tain 16  en.  ft.;  a  piece  16  ft.  long,  2]  i't.  broad,  and  1  ft.  thick  must 
contain  2J  times  16  cu.  ft.,  or  40  cu.  ft. ;  and  a  piece  16  ft.  long,  2\  ft. 
broad,  and  1]  ft.  thick  must  contain  1]  times  40  cu.  ft.,  or  50  cu.  ft. 

2.  A  rectangular  piece  of  timber  whose  cubic  contents 
are  50  cubic  feet  is  16  feet  long  and  2J  feet  broad. 
What  is  its  thickness? 

16  x  21  =A0.  Solution.— The  cubic  contents,  50  cubic  feet, 

are  the  product  of  the  length,  breadth,  and  thick- 
50+40=11      ness. 

The  thickness,  then,  must  be  the  quotient  arising  from  the  division 
of  the  cubic  contents,  50  cu.  ft.,  by  the  product  of  the  given  dimen- 
sions, 16  and  21,  =40,  which  gives  1\  ft.  as  the  thickness  required. 

3.  How  many  cords  of  4-foot  wood  in  a  range  28  feet 
in  length  and  6  feet  in  height? 

Solution. — The  contents 
28x4x6  =  672,  no.  ofcu.ft.    of  the  range  eqnal  28  x  4 

672+    16     =    42,  no.  of  cd.  ft.     x  6  =  672  cubic  feet. 

42+     8     =5{,  no.  of  cords.         Since  16  cu-  ft-  are  *  cd- 

ft.,  there  must  be  as  many 
cd.  ft.  as  there  are  times  16  cu.  ft.  in  672  cu.  ft.,  or  42  cd.  ft. 

Since  8  cd.  ft.  are  1  cord,  there  must  be  as  many  cords  as  there  are 
times  8  cd.  ft.  in  42  cd.  ft.,  or  5J  cords. 

4.  How  many  cord  feet  are  in  a  load  of  wood  8  ft. 
long,  3  ft.  6  in.  wide,  and  4  ft.  high? 

451.  Rules  for  Measurement  of  Rectangular  Solids.— 1.  Mul- 
tiply the  length,  width,  and  thickness  together, 
and  the  product  will  denote  the  cubic  contents. 

2.  Divide  the  cubic  contents  by  the  product  of  any 
two  of  the  dimensions,  and  the  quotient  will  denote 
the  other  dimension. 


206  APPEXDfX. 

PMOBLMMS4 
452. — 1.  A  bin  is  8  feet  long,  6  feet  broad,  and  4  feet 
(j  inches  deep.     How  many  cubic  feet  is  its  capacity  ? 

2.  A  load  of  wood  8  feet  long  and  3  feet  wide  is 
piled  5  feet  4  inches  high.  How  many  cords  are  in  the 
load? 

3.  A  rectangular  wall  is  120  feet  long,  2J  feet  wide, 
and  5  feet  high.     How  many  cubic  feet  does  it  contain  ? 

4.  How  many  cubic  feet  of  earth  must  be  removed  in 
excavating  a  cellar  30  feet  long,  22  feet  wide,  and  8  feet 
deep? 

5.  A  rectangular  block  of  marble  containing  36  cubic 
feet  is  8  feet  long  and  2  feet  3  inches  thick.  What  is 
its  width  ? 

6.  How  much  will  it  cost  to  remove  a  range  of  4-foot 
wood  which  is  48  feet  long  and  6  feet  high,  at  80  cents 
a  cord  ? 

Capacity  of  Bins,  Vats  and  Cisterns. 

453.  The  Standard  Bushel  in  the  United  States  con- 
tains 2150.42  cubic  inches. 

Hence,  a  cubic  foot,  expressed  in  bushels,  is  equivalent  to  ^xW-is 
=  .8036,  or  about  .8  of  a  bushel. 

454.  The   Standard  Gallon,   liquid    measure,    contains 

231  cubic  inches. 

Hence,  a  cubic  foot,  expressed  in  gallons,  is  equivalent  to  l^f  = 
7.48,  or  about  Tk  gallons. 

455.  A  Bin  for  coal  will  hold  a  ton  (2000  pounds) — 

Of  Lehigh  coal  for  every  36  cubic  feet. 
Of  Lackawanna  coal  "       38     "        " 
Of  Franklin  "      "       40     "        " 


APPENDIX.  207 

Wit  I TTEN  EXERCISES. 

456e — I,  How  many  bushels  of  corn  can  be  put  into 
a  bin  8  feet  long,  5  feet  broad,  and  5  feet  deep? 

SOLUTION. 

8x5x5=  £00,  number  of  cubic  feet 
200  x.8  —  160,  number  of  bushels. 

2.  A  rectangular  cistern  is  15  feet  long,  10  feet  broad, 
and  8  feet  deep.  How  many  hogsheads,  of  63  gallons 
each,  will  it  hold  ? 

SOLUTION. 

15  x  10  x  8  =  1200,  number  of  cubic  feet 
1200  x     7\       =  9000,  number  of  gallons. 
9000  +  63         —  14>2j,  number  of  hogsheads. 

3.  I  have  a  bin  8  feet  long,  6  feet  broad,  and  5  feet 
deep.     How  many  tons  of  Franklin  coal  will  it  hold? 

4.  A  bin  will  exactly  contain  160  bushels  of  wheat. 
What  is  its  capacity  in  cubic  feet? 

5.  I  have  a  cistern  6  feet  long,  4  feet  wide,  and  5  feet 
deep.  How  many  hogsheads,  of  63  gallons  each,  is  its 
capacity  ? 

6.  A  vat  will  hold  exactly  3840  gallons.  What  is 
its  capacity  in  cubic  feet  ? 

457.  Rules  for  Estimating  Bins,  Vats,  or  Cisterns.— 1.  Mul- 
tiply the  contents  in  cubic  feet  by  .8  for  bushels,  or 
by  7*  for  gallons. 

2.  Divide  the  raparih/  in  bushels  by  .8,  or  in 
gallons  by  7j,  for  contents  in  cubic  feet. 


208  APPENDIX. 

PROBLEMS. 

458. — 1.  How  many  bushels  of  grain  will  fill  a 
cubical  box  whose  dimensions  are  each  5  feet? 

2.  A  vat  has  96  cubic  feet  of  interior  space.  How 
many  gallons  of  water  will  it  hold  ? 

3.  A  tank  is  6  feet  long,  4  feet  6  inches  broad,  and  5 
feet  deep.  How  many  pounds  of  water  will  it  contain, 
the  weight  of  a  gallon  of  water  being  8J  pounds? 

4.  A  chest  will  contain  exactly  100  bushels  of  grain. 
What  are  its  contents  in  cubic  feet? 

5.  A  wagon-body  is  8  feet  long,  3  feet  6  inches  wide, 
and  2  feet  deep.     How  many  bushels  will  it  contain? 

6.  A  bin  is  6  feet  long  and  4  feet  wide.  How  deep 
must  it  be  to  contain  exactly  2  tons  of  Lehigh  coal  ? 

7.  I  have  a  bin  which  exactly  holds  5  tons  of  Lack- 
awanna coal.  It  is  10  feet  long  and  4  feet  deep.  What 
is  its  width  ? 

8.  How  many  bushels  of  wheat  can  be  put  into  a  bin 
that  is  8  feet  long,  6  feet  3  inches  wide,  and  4  feet  6 
inches  deep  ? 

9.  I  have  a  bin  6  feet  long,  6  feet  wide,  and  6  feet 
deep,  and  two  others  each  3  feet  long,  3  feet  wide,  and  3 
feet  deep.  How  many  more  bushels  will  the  first  hold 
than  the  other  two  ? 


459.  Test  Questions.— 1.  What  is  a  line?  A  straight  line? 
Kn  angle?  A  perpendicular  line?  A  right  angle?  A  rectangle? 
A  rectangular  solid? 

2.  What  are  the  dimensions  of  a  rectangular  surface?  The  area 
of  a  rectangle?  What  are  the  dimensions  of  a  rectangular  solid? 
The  cubic  contents  of  a  rectangular  solid? 

3.  How  many  cubic  inches  does  the  standard  bushel  contain  ?  The 
standard  gallon  ? 


APPENDIX.  209 

MISCELLANEOUS  PROBLEMS. 

460.  The  Articles  in  parentheses  denote  the  portions 
of  the  text  for  which  the  problems  may  be  used  as  sup- 
plementary exercises. 

(Art.  73.) 

1.  Express  by  figures,  fifty-nine  thousand  nine; 
seven  thousand  eighty;  and  fifty-one  thousand  one 
hundred  three. 

2.  Write  and  read  134640;  60041 ;  and  4602000. 

3.  What  is  the  sum  of  two  hundred  fifty-two  thou- 
sand six  hundred  six  and  one  hundred  seventy-two 
thousand  nine  hundred  seventy-five? 

4.  How  many  are  96004  —  964?  83334  —  9453? 

5.  If  you  have  5675  dollars,  and  should  spend  4987, 
how7  much  would  you  have  left  ? 

6.  How  many  are  7806  +  760  +  9376  +  97  ? 

7.  How  many  are  95631  —  777  added  to  66406 -f 
9972? 

8.  A  mill  had  6750  barrels  of  flour.  To  A  it  sold 
2173  barrels,  and  to  B  978  barrels.  How  much  then 
had  it  unsold  ? 

9.  How  many  are  152445  +  707050  less  93987? 

10.  The  coach  was  first  made  in  England  in  the  year 
1564,  which  was  1083  years  after  iron  shoes  were  first 
made  for  horses.  In  what  year  were  the  iron  shoes  first 
made? 

11.  How  many  are  1685  +  75  +  832  +  9675  ? 

12.  How  many  are  59  +  641  +  9  +  8086  +  93015  ? 

13.  How  many  are  19678  —  3789,  less  10000  — 
9889? 


210  APPENDIX. 

14.  A  man  had  50000  dollars,  and  paid  from  it  for 
a  house  10550  dollars,  for  land  18075,  and  for  goods 
15787.     How  much  money  had  he  then  left? 

(Art.  117.) 

1.  How  much  will  15  horses  cost  at  225  dollars  each? 

2.  Add  3605  to  the  product  of  716  by  97. 

3.  Subtract  1148  from  the  product  of  517  by  68. 

4.  A  farmer  bought  316  sheep  at  3  dollars  each,  and 
sold  250  of  them  for  1 000  dollars,  and  the  remainder  at 
a  loss  of  a  dollar  each.     How  much  did  he  make? 

5.  From  1840  x  18  take  2045  x  13. 

6.  How  much  is  13465  divided  by  47? 

7.  A  planter  has  64575  gallons  of  molasses.  How 
many  casks  of  63  gallons  each  will  be  required  to 
hold  it? 

8.  How  many  are  225  x  36,  divided  by  810-*- 15? 

9.  When  64  acres  of  land  can  be  bought  for  1600 
dollars,  how  many  acres  can  be  bought  for  3225  dollars? 

10.  If  the  multiplier  is  47  and  the  product  is  13113, 
what  is  the  multiplicand  ? 

11.  If  the  divisor  is  71  and  the  quotient  1002,  what 
is  the  dividend  ? 

12.  The  quotient  is  1365,  the  divisor  63,  and  the  re- 
mainder 14.     What  is  the  dividend? 

13.  What  is  the  quotient  of  13675  divided  by  18? 
By  63?     By  37? 

14.  What  is  the  quotient  of  67350  divided  by  100? 
By  31?     By  60? 

15.  Bought  224  barrels  of  flour  for  1568  dollars,  and 
sold  the  same  for  2128  dollars.  What  was  the  gain  on 
each  barrel  ? 


APPENDIX.  211 

(Art.  238.) 

1.  Reduce  14-j-^-  to  an  improper  fraction. 

2.  What  is  the  Value  of  Vs5  ?     °f  "H"? 

3.  How  many  ones  are  there  in  ^-J5-?     In  6Ier3  ?     In 

1119 

tt  •      • 

4.  Change  §-,  -§,  and  T7^  each  to  twenty-fourths. 

5.  Arrange  |-,  -^-,  and  -|  in  the  order  of  their  values. 

6.  John  is  llf  years  old,  and  William  13^.  What 
is  the  sum  of  their  ages? 

7.  A  pole  stands  -f\  of  its  length  in  the  mud,  -^  of 
its  length  in  the  water,  and  the  rest  above  water.  How 
much  of  its  length  is  above  water  ? 

8.  What  is  the  difference  between  -|  and  |  ? 

9.  Wrhat  is  the  difference  between  y  and  5§  ? 

10.  How  much  greater  is  the  sum  of  3^  and  2|  than 
their  difference? 

11.  What  is  the  difference  between  18  and  a  fifth  of 
fifteen-fifths? 

12.  At  f  of  a  dollar  a  bushel,  what  will  84  bushels 
of  potatoes  cost? 

13.  If  a  boy  can  earn  \\  dollars  in  a  week,  how 
much  can  he  earn  in  -|  of  a  week  ? 

14.  At  2-J-  dollars  a  day,  in  how  many  days  will  a 
man  earn  28f  dollars? 

15.  How  much  is  §  of  ff  ?  \  of  -i-f2-? 

16.  What  number  added  to  -f  will  give  \\\ ? 

17.  If  5  yards  of  cloth  cost  If  dollars,  what  part  of 
a  dollar  is  it  a  yard  ? 

18.  What  fraction  is  equivalent  to  J  of  f  of  f? 

19.  A  man  owning  f  of  a  farm  sells  f  of  his  share 
for  450  dollars.  At  this  rate  what  is  the  value  of  the 
whole  farm? 


212  APPENDIX. 

20.  A  certain  estate  is  worth  24000  dollars,  and  1^ 
of  the  value  of  the  estate  is  j-  of  the  value  of  the  house 
upon  it.     What  is  the  value  of  the  house? 

21.  A  farmer  has  36f  bushels  of  corn  in  a  bin,  which 
is  just  f  of  all  he  raised.     How  much  did  he  raise? 

22.  Divide  2  by  the  sum  of  2§  and  f,  and  to  the 
quotient  add  If  —  f. 

(Art.  384.) 

1.  How  many  dollars  will  112  bushels  of  apples  cost 
at  62  J  cents  a  bushel? 

2.  Bought  2  bushels  of  chestnuts  at  $3.50  per  bushel, 
and  sold  them  at  8  cents  a  pint.  How  much  was 
gained  by  the  transaction  ? 

3.  A  man  is  31  years  old,  allowing  365J  days  to  a 
year.     How  many  hours  has  he  lived? 

4.  The  distance  between  two  places  is  exactly  15 \ 
miles.     How  many  yards  are  they  apart? 

5.  A  lot  of  land  containing  exactly  47  square  rods 
was  sold  at  2  cents  a  square  foot.  How  much  did  it 
sell  for? 

6.  How  many  acres  are  4392  square  rods? 

7.  How  many  hogsheads  of  vinegar  of  63  gallons 
each,  at  5  cents  a  quart,  can  be  bought  for  $75.60? 

8.  When  hay  is  $25  per  ton,  how  much  is  it  per  hun- 
dred-weight? 

9.  How  many  ounces  are  there  in  5  tons? 

10.  How  many  acres  are  25090560  square  inches? 

11.  How  many  pounds  are  37440  troy  grains? 

12.  How  many  chests  of  tea  of  35  pounds  each,  at 
45  cents  a  pound,  can  be  bought  for  $63  ? 


APPENDIX.  213 

(Art.  373.) 

1.  How  many  minutes  are  there  in  1  week  4  days  15 
hours  ? 

2.  How  much  is  the  sum  of  14  lb.  3  oz.  11  pwt. ; 
5  lb.  7  oz.  13  pwt.,  and  11  oz.  17  pwt,  13  gr.? 

3.  From  59°  42'  15"  take  11°  39'  47". 

4.  How  many  ounces  are  there  in  13  cwt,  67  lb. 
14  oz.? 

5.  How  much  must  be  paid  for  183073  square  yards 
of  land  at  50  cents  a  square  rod? 

6.  A  certain  range  of  wood  contains  exactly  10  cords. 
If  2  loads,  each  containing  1  cord  3  cord  feet,  be  taken 
from  it,  how  much  will  be  left?  What  will  the  remain- 
der be  worth  at  75  cents  per  cord  foot? 

7.  What  is  the  quotient  of  34  cd.  6  cd.  ft.  4  cu.  ft. 
divided  by  4  ?     Of  118  bu.  1  pk.  5  qt.  -*-  6  ? 

8.  What  is  the  product  of  10  reams  5  quires  13 
sheets  multiplied  by  9?  13  A.  15  p.  3  sq.  yd.  x  10? 

9.  Three  thieves  carry  off  from  a  house  7  silver  cups, 
each  weighing  10  oz. ;  1  dozen  and  9  silver  forks,  each 
weighing  2  oz.  8  pwt, ;  and  13  silver  spoons,  each 
weighing  3  oz.  Dividing  the  silver  among  the  thieves, 
and  giving  to  one  of  them  a  double  share,  how  much 
would  each  have? 

10.  What  is  the  value  of  1  mi.  25  rd.  2  yd.  2  ft. +  5 
mi.  150  rd.  3  yd.  2  ft,? 

11.  What  is  the  value  of  71  gal.  3  qt,  1  pt.  3  gi.  x  8? 
Of  68  d.  18  h.  56  min.-4? 

12.  A  dray-load  is  found  to  weigh  1  ton  4  cwt,  15 
lb.,  and  it  consists  of  21  boxes.  What  is  the  weight  of 
each  box? 


214  APPENDIX. 

(Art,  407.) 

1.  Write  one  thousand,  and  one  ten  thousandth. 

2.  Write  fifteen,  and  fifteen  millionths. 

3.  Find  the  sum  of  eleven  and  six  hundredths,  added 
to  thirteen,  and  one  hundred  six  thousandths. 

4.  From  93  take  .0093;  from  1  take  .000001. 

5.  What  is  the  product  of  1.04  multiplied  by  .104? 

6.  Express  .275  as  a  common  fraction  in  its  simplest 
form. 

7.  Express  T^  as  a  decimal  of  3  orders. 

8.  What  is  the  value  of  17  divided  by  1.7? 

9.  What  is  the  value  of  (-  x  '-z\  x  1.3? 

10.  Divide  1.56  by  .005,  and  .0003  by  .05,  and  find 
the  sum. 

11.  Multiply  7  ten-thousandths  by  15  thousandths, 
and  divide  the  result  by  .25. 

12.  How  many  thousand  feet  of  boards  at  $13.50  a 
thousand  can  be  bought  for  $101.25? 

13.  A  man  traveled  5  days;  the  first  day  he  went 
16.05  miles,  the  second  35.16  miles,  the  third  21y^7 
miles,  the  fourth  11.009  miles,  and  the  fifth  31^^- 
miles.     How  far  did  he  travel  in  all  ? 

14.  I  bought  a  cask  of  refined  petroleum  containing 
48.5  gallons.  How  much  of  it  can  I  sell  and  have  left 
13.125  gallons? 

15.  A  person  sold  .15  of  an  estate  to  one  person,  and 
then  J  of  the  remainder  to  another  person.  What  part 
of  the  estate  did  he  still  retain?  r^j 


APPENDIX.  215 

(Art,  433.) 

1.  How  much  is  5  J  %  of  8670   yards?    11  %  of 
$125  ?     15  per  cent,  of  624  bushels  ? 

2.  What  %  is  made  by  selling  goods  at  $76.56  which 
cost  $63  ? 

3.  A  horse  which  cost  $250  was  sold  at  $75  above 
cost.     What  was  the  rate  %  of  gain  ? 

4.  Bought   goods    for    $1250    and    sold    them    for 
$1206.25.     What  was  the  loss  per  cent,? 

5.  Bought  coal  at  $7.50  per  ton,  and  sold  it  so  as  to 
gain  15  %.     At  what  price  was  it  sold? 

6.  What  is  the  interest  of  $245  for  3  years,  at  7  %  ? 
For  2  years  6  months,  at  7  %  ? 

7.  What  is  the  amount  of  $370  for  8  months  18  days, 
at  9  %? 

8.  What  is  the  interest  of  $1000  for  1  month  6  days, 
at  8  %? 

9.  What    is    the   amount    of   $1250    for    2   years   2 
months  15  days,  at  6  %  ? 

10.  What   is    the    amount    of   $965    for   3  years    6 
months  24  days,  at  4  %  ? 

11.  What   is   the   interest   of  $50.50   from   Jan.  1, 
1877,  to  Oct,  10,  1878,  at  7  %? 

^     12.  What  is  the  amount  of  $834.80  from  Feb.   15, 
1877,  to  Nov.  21,  1878,  at  6  %  ? 

13.  What  is  the  interest  of  $500  from  July  1,  1877, 
to  Jan.  16,  1878,  at  7  %  ? 

14.  What  is  the  amount  of  $1540.50  from  April  1  I 
to  Oct,  21,  1877,  at  8  %? 

15.  What  is  the  amount  of  $2000  from  May  3,  1877, 
to  June  13,  1878,  at  5  %? 


ANSWERS. 


Art.  42. 

Art.  49. 

6.  19497. 

3.  164. 

1.  9602. 

Art.  69. 

4.  116. 

2.  14152. 

1.  42169. 

5.  96. 

3.  425435. 

2.  5700. 

6.  169. 

4.  870  yards. 

3.  4785. 

9.  88. 

5.  9514  pounds. 

4.  1858. 

10.  137. 

6.  2296  men. 

5.  3630. 

11.  149. 

7.  1834  cords. 

6.  10375. 

12.  88. 

8.  9976. 

7.  2195. 

Art.  43. 

9.  4084. 

8.  16911. 

1.  208. 

2.  168. 

10.  25308. 

11.  1597124. 

Art.  70. 

1.  61. 

3.  147. 

Art.  50. 

2.  3110  dollars. 

4.  209. 

1.  1799. 

3.  811. 

5.  178. 

2.  6125  dollars. 

4.  3268  feet. 

6.  220. 

3.  1045. 

5.  1785  dollars. 

7.  159. 

4.  10196. 

6.  910;  989. 

8.  179. 

5.  5649. 

7.  18981. 

9.  194. 

6.  573  dollars. 

8.  127987. 

10.  158. 

Art.  51. 

Art.  73. 

11.  148. 

1.  204. 

1.  2340. 

12.  178. 

2.  281. 

2.  3334. 

Art.  45. 

3.  427. 

3.  834. 

2.  231. 

4.  2068. 

4.  4110. 

3.  570. 

5.  15155. 

5.  11567. 

4.  1041. 

6.  157. 

6.  7457. 

5.  963. 

7.  341. 

8.  2275. 

6.  488. 

8.  544. 

9.  3572. 

Art.  47. 

1.  8660. 

9.  2465. 
10.  7606. 

10.  4925. 
Art.  86. 

2.  7886. 

Art.  66. 

2.  1250.  * 

3.  3885. 

2.  1716. 

3.  5360. 

4.  5942. 

3.  4899. 

4.  2660. 

6.  2722. 

4.  5594. 

5.  3100. 

Art.  48. 

6.  7099. 

6.  31000. 

1.  903  dollars. 

7.  8902. 

7.  86800. 

2.  806  men. 

8.  5200. 

Art.  88. 

3.  881  miles. 

Art.  68. 

2.  2996. 

4.  921  bushels. 

1.  918. 

3.  14761. 

5.  1301  feet. 

2.  538. 

4.  1824. 

6.  7907  tons. 

3.  18888. 

5.  9512. 

7.  9919  bales. 

4.  81907. 

6.  14652. 

8.  1279  horses. 

6.  1001. 

7.  7140. 

216 

AN8  WEMS. 


217 


Art.  90. 

1.  7506  dollars. 

2.  3612  hogsheads. 

3.  7259  yards. 

4.  36081  bushels. 

5.  9855  days. 

6.  2494  miles. 

7.  731000  tons. 

8.  376800  acres. 

Art.  91. 

2.  158632. 

3.  2415207. 

4.  1010025. 

5.  22612390. 

6.  13500  dollars. 

7.  21400. 

8.  210511. 

Art.  92. 

1.  4100  dollars. 

2.  13104. 

3.  117600. 

4.  293760. 

5.  12769. 

6.  277008. 

7.  146520. 

8.  154350. 

9.  2199582. 
10.  57600000. 

Art.  110. 

2.  108. 

3.  448. 

4.  137. 

5.  141. 

6.  143. 

7.  131. 

8.  117. 

9.  398. 
10.  72. 

Art.  113. 

1.  681J. 

2.  57 5f. 

3.  lOOf. 

4.  515. 

5.  332^. 

6.  260}$. 

7.  21*. 

8.  558TV 


Art 
1. 
2. 


,113. 

182|  apples. 
187f  cents. 
133|feet. 
59  f  rods. 
691  dollars. 


4. 
5. 

6.  578T5T  days. 

8. 

9. 

10. 


Art 

1. 
2. 
3. 
4. 
5. 
6. 


8. 


210}i  pounds. 
2557  dollars. 
320  cents. 
483|. 
.114. 
358TV 
406  dollars. 
49. 
62. 
41. 
62,  and  7  gallons 

will  remain. 
36,  and  5  acres 

will  remain. 
57,  and  67  feet 

will  remain. 

Art.  117. 

1.  25  dollars. 

2.  9443C, 

3.  1012  dollars. 

4.  910. 

5.  4350  dollars. 

6.  191T82  dollars. 

7.  262  bu.  of  oats, 

393     bu.     of 
wheat. 

8.  54J8-. 

9.  1775  days. 

10.  4681  dollars. 

11.  1800  dollars. 

12.  7f«f. 

13.  51. 
Art.  126. 

2.  2,  2,  3,  7. 

3.  3,  5,  5. 

4.  2,  2,  2,  2,  2,  3. 

Art.  128. 

1.  5,  19. 

2.  3,  3,  7. 

3.  2,  2,  23. 

4.  2,  61. 


11. 
12. 


5.  2,  2,  29. 

6.  2,  2,  2,  23. 

7.  2,  2,  3,  3,  3. 

8.  2,  2,  2,  5,  5. 

9.  2,  2,  2,  91. 
10.  2,  2,  37. 

2,  3,  5,  7. 

3,  5,  7,  7. 

Art.  134. 

2.  9. 

3.  7. 

4.  7. 

Art.  144. 

1.  144. 

2.  108. 

3.  168. 

4.  42. 

5.  126. 

6.  57. 

7.  120. 

8.  240. 

9.  560. 
10.  60. 

Art.  148. 

3.  25|. 

4.  5,  4. 

Art.  150. 

1.  2f 

2.  5. 

3.  15. 

4.  4f. 

5.  2. 

6.  6. 

7.  27. 

8.  5. 

10.  6f. 

11.  19. 

12.  m . 

Art.  108. 

2.  V;  ¥o°. 

3.  a$*;*H 

5.  V;  W- 

6.  V;  W- 

Art.  170. 

1.      y  . 

2       6* 


218 

ANS  WEBS. 

3.  1JL6. 

Art.  180. 

11.  63f. 

4.  if* 

Lf 

12.  20J-J, 

5.  Vs1 

2.  J. 

Art.  201. 

6.  W- 

3.  |. 

9     -7- 

^'     28' 

7.  H** 

4.  tV 

3.  i%. 

8.  w- 

5.  TV 

5.  7,V 

9.  W- 

6.  f 

6.  19A. 

10.  4*, 

7.  i. 

Art.  303. 

11        209 
J.i.     15  . 

8.  J. 

1         5 
!•    2¥- 

12.  \%6. 

Art.  180. 

O        3 
<*•    T8- 

Art.  172. 

.Q      55.    5  4 
°'    66  )   ^6* 
A        6.4. 
4=.    T8)    T8) 

Q         9 

«•     62* 

4.  it. 

2.  91. 

18* 

3.  73. 

Art.  102. 

5.  tf  j. 

5.  13T\. 

1«    T2)     12* 

o.  34. 

6.  I7&. 

Art.  174. 

1.  14. 

O       1  8  .    _3_  . 
*■••     3  0)     30) 

A- 

7.       S/q. 

O      48.20, 
«5»     8  8  )     8  8  ) 
J.      24.16. 
*='     3  6)     36) 

14 

8  8- 

8.  8J. 

9.  66^. 

2.  26f. 

3.  17. 

K      12.     20. 
°«     24)     24) 

A- 

10.  42T\. 

ft       16.35. 
«.    ?0 )    40 > 

li- 

11. 2&. 

4.  80J. 

rj      20.21. 
*•    TO  )   70 ) 

ft- 

Art.  207. 

5.  16|. 

8.    ?2)     42) 

1  6 
T2- 

3.  2J;  7TVT;  5Jfe 

6    25f. 

7.  79. 

8.  31. 

Q      32.    35. 
«*•    ¥o  )   ¥o  ) 

if 

4.  414. 

in     s  .    9  .  ip_ 

J-U.     12-)    T2  )     12- 
11         2  4.110.    153 
L*-    T'g'O  )     180  )     18  0* 

Art.  200. 

1.  6TV 

9.  24|f. 

lO       495.     180.     336 
1<5.    Y2  0)    72  0)    72  0* 

2.  16TV 

10.  21. 

Art.  106. 

3.  19*. 

11.  304J. 

2.  2& 

4.  li. 

12.  72/7. 

3.  2& 

5.  4f. 

13.  113;  78  J  yards. 

5.  2Uf 

6.  2f. 

Art.  178. 

6.  2tf. 

7.  2i 

O       42.12 
*'    1~8>     6T- 

8.  33J. 

8.  3ff. 

3.  If;  f*. 

Art.  108. 

9.  60f. 

Art.  180. 

1.  2^V. 

10.  104. 

1.  f|. 

2.    t¥2. 

11.  129^. 

2.  4». 

3.  Hi. 

12.  1215. 

q     45 

4.  |g. 

4.  Wo- 

Art.  214. 

5.  6ft 

3.  30#. 

5.  &. 

6.  ioa. 

4.  49J. 

6.  T3oV 

7.  40^. 

5.  25  r57. 

Art.  184. 

2.  J;  A- 

8.  2|f. 

9.  3§. 

6.  280. 

7.  102TV 

3.  A;  tV 

10.  71. 

8.  387. 

-4iVS  VKEJJ& 

9.  78. 

5.  867. 

5.  4f. 

Art.  316. 

6.  49f. 

6.  15f  dollars. 

3.1f. 

7.  238|-. 

7.  4|. 

4.  36*;  181ft. 

8.  3371. 

8.  TV 

Art.  318. 

9.  84. 

9.  14f. 

1.  ft. 

10.  40*. 

10.  640. 

2-  t¥i- 

11.  81. 

Art.  253. 

3.  ft. 

12.  200. 

3.  $16.25 

4.  |. 

14.  6;  17 A;  6ft. 

4.  $82.28 

5.  ftft- 

15.  10. 

6.  $262.37 

6.  2}|f. 

Art.  331. 

7.  $7015.50 

7.  10H- 

2.  1ft. 

Art.  255. 

8.  2ft 

3.  81. 

2.  $77.06 

9.  lfH. 

Art.  333. 

3.  $77,915 

io.  H- 

1.  4. 

4.  $26.98 

11.  20. 

2.  1ft. 

5.  $103.81 

12.  89f, 

3.  1ft. 

6.  $3688.18 

13.  m- 

4.  1ft- 

7.  $83.50 

14.  J. 

5.  2*. 

Art.  257. 

15.1. 

10.  5. 

2.  $490.72 

16.  2|  dollars. 

11.  If 

3.  $447.15 

Art.  333. 

12.  }. 

4.  $78,125 

Q        3    .        1      .    Q  9 

14.  6. 

5.  $228.72 

Art.  334. 

15.  4f. 

6.  $260,169 

1.  ft- 

16.  4f ;  41. 

7.  $9356. 

2.  ft. 

17.  31. 

8.  $13065. 

3.  ft. 

Art.  336. 

9.  $180576. 

4.  ft. 

l.  H3-- 

10.  $34055. 

5.  ft. 

O       14.15 

11.  $10303. 

6.  ft. 

3.  1;  1. 

12.  $391.80 

7.  2ft. 

4.  14|  dollars. 

13.  $17503.50 

8.  2H. 

5.|*. 

14.  $1510.44 

9.  11!- 

6.  261;  311. 

15.  $76800. 

10.  8ft  dollars. 

7-  2ft;  2|f 

Art.  259. 

Art.  336. 

8.  5ft;  5}f. 

3.  $40.65 

2.  73f. 

9.  21  tons. 

4.  $6.10 

3.  331;  134|;  2201. 

10.  161  dollars. 

5.  $25.06 

Art.  338. 

Art.  338. 

6.  $6.12 

1.  26|. 

1.  26  yards. 

7.  $93.56 

2.  911. 

2.  If;  *Hti  4. 

8.  $1005. 

3.  52J. 

3.  11. 

9.  $3.64 

4.  64. 

4.  126  dollars. 

10.  $634,055 

219 


220                 ANSWIMS. 

11.  $56.75 

8.  $450. 

12.  $1.31 

Art.  334. 

13.  $19.04 

1.  527040. 

14.  $1111.44 

2.  $165. 

15.  $16.75 

3.  12. 

17.  20. 

6.  $403.20 

18.  $5.25 

7.  23040. 

19.  85. 

8.  40. 

20.  $2641. 

9.  40. 

21.  $12. 

Art.  341. 

Art.  262. 

2.  175  in. 

2.  $90. 

3.  7515  sq.  in. 

3.  $63. 

4.  $8. 
Art.  264. 

1.  $947,50 

2.  $66.75 

3.  $8510. 

4.  $32. 
Art.  270. 

Art.  343. 

1.  11056//. 

2.  18755  sq.  yd. 

3.  7569  ft. 

4.  4700260  cu.  in. 

5.  253  qt. 

6.  4601  gr. 

7.  142544  oz. 

Bill  receipted,  $21.37J. 

8.  335  qt. 

Art.  277. 

9.  $300. 

5.  15;  99. 

10.  $504. 

Art.  287. 

11.  $44. 

9.  155520. 

12.  441300  h. 

10.  36. 

Art.  345. 

11.  65. 

2.  4  yd.  2  ft.  7  in. 

Art.  300. 

3.  5  sq.  yd.  7  sq.  ft.  27  sq.  in. 

5.  $30.24 

4.  10  h.  17  m.  18  sec. 

6.  $9.60 

Art.  347. 

Art.  303. 

1.  3°  4/ 16". 

6.  6. 

2.  3  A.  140  P. 

Art.  312. 

3.  1  mi.  138  rd.  4  yd. 

7.  480. 

4.  100  cu.  yd.  20  cu.  ft.  10( 

Art.  323. 

5.  2400. 

6.  72. 

7.  12600. 

cu.in. 
5.  8  bu.  2  pk.  3  qt. 

6.  9  oz.  11  pwt.  17  gr. 

7.  4  T.  9  cwt.  5  lb.  14  oz. 

8.  1  hhd.  20  gal.  3  qt. 

Art.  327. 

9.  1A.  40  sq.  rd. 

8.  366. 

10.  lmi.  100  rd. 

9.  5785560. 

11.  12  63  23. 

Art.  331. 

12.  50  y. 

4.  2544. 

Art.  351. 

5.  $51.60 

2.  14  mi.  17  rd. 

6.  24. 

3.  23  gal.  2  qt. 

7.  $72. 

ANSWERS.                                            221 

Art.  353. 

3.  10  lb.  1  oz.  10  pwt.  17  gr. 

1.  19cwt.  lib.  5oz. 

Art.  369. 

2.  87  A.  76  P. 

1.  lhhd.  6  gal.  1  qt.  1  pt.  2f  gi. 

3.  20  cu.  vd.  21  cu.  ft.  28  cu.  in.  i 

2.  50  cu.  ft.  1121  cu.  in. 

4.  lib.  9oz.  19pwt.  6gr. 

3.  3  A.  95  P.  22  sq.  yd.  8^  sq.ft. 

5.  191  bu.  3pk. 

4.  4  bu.  0  pk.  4  qt.  1  pt.  \\  gi. 

6.  54  rd.  2  vd.  2  ft. 

5.  3  cd.  2  cd.  ft.  5  cu.  ft.  576 

7.  15  d.  23  h.  37  m. 

cu.  in. 

8.  52°  55'  48". 

6.  8y.  Omo.  18  d. 

9.  l'bu.  lpk.  3qt.  1  pt. 

7.  1  cwt.  3  lb.  14$  oz. 

10.  33  A.  lOOsq.rd. 

8.  28  mi.  4  rd. 

11.  47  T.  8cwt.  561b. 

9.  40  A.  35  P. 

Art.  356. 

10.  1  oz.  5  pwt.  12|  gr. 

2.  31b.  10  oz.  2pwt. 

11.  35  bags. 

3.  1  gal.  2  qt.  1  pt. 

12.  1  A.  61  P.   10  sq.  yd.  108 

4.  8  yd.  2  qr.  3  na. 

sq.  in. 

Art.  358. 

13.  7  h.  45  min.  50  sec. 

1.  8  m.  300  rd.  1  yd. 

Art.  372. 

2.  2  A.  155  P.  5sq.  yd. 

1.  1793  steps. 

3.  1  wk.  4  d.  23  h. 

2.  1  mi.  6  rd. 

4.  22  T.  19cwt.  941b.  lloz. 

3.  28  lots. 

5.  3g  53  19. 

4.  63000". 

6.  1°  51'  15". 

5.  2  \  cords. 

7.  19  m.  34  rd.  2  yd.  1ft.  6  in. 

6.  16  spoons;  and  16  pwt.  will 

8.  1  y.  10  mo. 

remain. 

9.  28  gal.  1  pt. 

7.  15  days. 

10.  16°  34'  17". 

8.  $43.20 

12.  52  y.  1  mo.  18  d. 

9.  $203.40 

Art.  362. 

10.  20f  loads. 

2.  32  rd.  0  yd.  2  ft. 

11.  9t%3t  hours. 

3.  811b.  5pwt.  16  gr. 

12.  10. 

Art.  364. 

Art.  379. 

L  8hhd.  12  gal.  1  qt. 

3.  Three   thousand    one    hun- 

2. 405  cu.  ft.  328  cu.  in. 

dred   sixty-five  ten  thou- 

3. 57  A.  92  P.  4  sq.  yd. 

sandths. 

4.  43  bu.  1  pk.  2  qt. 

4.  Five  hundred  fifteen,  and  six 

5.  64  wk.  6d.  4h. 

thousandths. 

6.  104°  V  52". 

6.  .1331 

7.  15  cd.  6  cd.  ft. 

7.  11011.011 

8.  9  cwt.  20  lb.  8  oz. 

Art.  381. 

9.  41  y.  3  mo. 

1.  Three  hundred  forty-seven 

10.  301  mi.  44  rd. 

thousandths. 

11.  281  A.  85  P. 

2.  One  thousand  four  hundred 

12.  11  hhd.  61  gal.  2  qt. 

five  ten-thousandths. 

13.  21b.  3oz.  11  pwt.  6gr. 

3.  Seventy-two   ten-thou- 

14. 336  A.  24  P. 

sandths. 

Art.  367. 

4.  One     hundred    thirty-eight 

2.  6rd.  2yd.  Oft.  9f  in. 

thousandth?. 

222 


ANSWERS. 


5.  Ten      thousand 

sixty-five  hun- 
dred thou- 
sandths. 

6.  Two  millionths. 

7.  Fourteen,      and 

five  thou- 

sandths. 

8.  Nine,  and  nine- 

ty-three thou- 
sandths. 

9.  One,     and     six 

thousand  four 
hundred  forty- 
four  ten-thou- 
sandths. 

10.  .5 

11.  .31 

12.  .04 

13.  .017 

14.  .434 

15.  .1111 

16.  14.05 
17%  9.003 

18.  1. 06444 

19.  .98 

20.  .125 

21.  .1018 

22.  6.75 

23.  2462.7 

24.  400.044 

25.  24.000024 

Art.  384. 

2.  80.928 

4.  22.44 

5.  4.257 

Art.  386. 

1.  16.275 

2.  41.151 

3.  1.182 

4.  34.039 

5.  8.505 

6.  .137  of  a  ton. 
-7.  97.02 

8.  121.875 

9.  114.028 
10.  $206. 


Art.  300. 

3.  2.865 

4.  .0450 

Art.  303. 

1.  17.5 

2.  1.75 

3.  17.5 

4.  1.75 

5.  3.01 

6.  .0301 

7.  30.1 

8.  30.1 

9.  4.872 

10.  .4872 

11.  4.872 

12.  48.72 

13.  7.5 

14.  215042. 

15.  3.65 

16.  $34.85 

Art.  305. 

3.  3.15 
1.34 

Art.  307. 

1.  1.23 

2.  43. 

3.  3.3 

4.  3.3833-f 

5.  .05 

6.  100. 

7.  .01 

8.  10. 

9.  .42 

10.  75. 

11.  10. 

12.  .123 

13.  .756 

14.  .03145 

15.  .0000905 

17.  .0609+ 

18.  15.5  yd. 

Art.  300. 

2.  &. 

3.  tI* 

4.  h 

5.  f. 


Art.  401. 

i.  B- 

2.  h 

3.  jV 

4.  tI*. 

5.  f. 

6.  i0: 

n     12 

»•    T2T- 

8.  tfe. 

9.  l\h 

Art.  403. 

2.  .75 

3.  .08 

Art.  405. 

1.  .85 

2.  .125 

3.  .025 

4.  .016 

5.  .625 

6.  .0625 

7.  .096 

8.  .008 

9.  1.275 
11.  .3846+ 

Art.  407. 

1.  *. 

2.  .0017 

3.  i. 

4.  .368+ 

5.  $55.25 

6.  31.856 

7.  20.4 

8.  $507.78 

9.  $60,477 

10.  5305. 

11.  100. 

12.  2146.85  pound* 

Art.  414. 

1.  .03 

2.  .04 

3.  .07 

4.  .09 

5.  .04J 

6.  .09£ 

7.  .17J 


A  XS  WEMS. 


223 


8.  .07| 

9.  .45 

10.  .67 

11.  1.25 

12.  1.35 

Art.  416. 

2.  43  pounds. 

3.  90  men. 

Art,  418. 

1.  $2.46 

2.  $34.09 

3.  11.6  yd. 

4.  132.5  rd. 

5.  160.3 

6.  27.625 

7.  106.78 

8.  105.6 

9.  $75. 

10.  36  men. 

11.  84cwt. 

12.  435  lbs. 

13.  $135. 

14.  $187.50 

15.  2176  bu. 

16.  11448. 

Art.  420. 

2.  5%. 

3.  m&. 


Art.  422. 

1.  16|%. 

2.  6f%. 

3.  6A#. 

4.  25%. 

5.  lfH%. 

6.  7%. 

7.  19%. 

8.  60%. 

9.  9ff%. 

10.  8%. 

11.  15%. 

12.  60%. 

13.  45%. 

14.  5%. 

15.  27|g|%. 

16.  40%. 

17.  20%. 
Art.  428. 

2.  $192.50 

3.  $265. 

4.  $168. 

5.  $191.67 

7.  $618.90 

8.  $727.56 

9.  $27,778 
10.  $12.60 

Art.  430. 
1.  $472.50 


2.  $3,105 

3.  41.777 

4.  $1236.667 

5.  $693,333 

6.  $5279.167 

7.  $18.36 

8.  $166,625 

9.  $.588 

10.  $813.61 

11.  $965.08 

12.  $300.90 

13.  $55,026 

14.  $1099.75 

15.  $1368. 

Art.  433. 

1.  $37.50 

2.  $6,681 

3.  $1.89 

4.  $5000. 

5.  $1095.85 

6.  $949,333 

7.  $12.60 

8.  10%. 

9.  $495. 

10.  $300. 

11.  $1854.16 

12.  $5205.493 


ANSWERS  TO  APPENDIX. 


Art.  444. 

Art.  452 

3.  76  ft. 

1.  216. 

4.  45. 

2.  1. 

Art.  446. 

3.  1500. 

1.   1400  sq.  rd. 

4.  5280. 

2.   1849  sq.  ft. 

5.  2. 

3.  43  rd. 

6.  $7.20 

4.  15  rd. 

Art.  456, 

5.  40  sq.  yd. 

3.  6. 

Art.  450. 

4.  200. 

4.  7. 

5.  14?, 

6. 

512. 

Ir 

t.  458. 

1. 

100. 

2. 

720. 

3. 

84373. 

4. 

125. 

5. 

44.8 

6. 

3  ft. 

7. 

4  ft.  9  in 

8. 

180. 

9. 

129.6 

224 


ANSWERS. 


ANSWERS  TO  MISCELLANEOUS  PROBLEMS. 


Art.  73. 

1.  59009. 

7080. 

51103. 

3.  425581. 

4.  95040. 
73881. 

5.  688  dollars. 

6.  18039. 

7.  171232. 

8.  3599  barrels. 

9.  765508. 

10.  481. 

11.  12267. 
11.   101810. 

13.  15778. 

14.  5588. 

Art.  117. 

1.  3375  dollars. 

2.  73057. 

3.  34008. 

4.  184  dollars. 

5.  6535. 

6.  286ff 

7.  1025. 

8.  150. 

9.  129. 

10.  279. 

11.  71142. 

12.  86009. 

f759tf. 

13.  ]217A. 
(  369|f. 

14.  \  2172U. 

I  1122JJ. 

15.  2J  dollars. 

Art.  238. 

1.  -1-79. 

2*.  26JJ;  64  V. 

3.  31;  63;  6° 

4.  1  fi   .      20  .      14' 

6.   25  &  years. 


v.  ih 

8.  TV 

9.  X. 

10.  V. 

11.  17f 

12.  $52.50 

13.  $3.00 

14.  11  J. 

1  K         21     .     91 

16.  L/o- 

17.  f. 

18.  I. 

19.  $1500. 

20.  $6300. 

21.  81fJ  bushels. 

22.  IJf. 

Art.  334. 

1.  $70. 

2.  $3.24 

3.  271746. 

4.  27280. 

5.  $255.9H 

6.  27/„. 

7.  6.  " 

8.  81.25 

9.  160000. 

10.  4. 

11.  6i. 

12.  4." 

Art.  372. 

1.  16740. 

2.  20  1b.lloz.lpwt. 

13  gr. 

3.  48°  2/  28". 

4.  21886. 

5.  83026. 

6.  843.50 

f8  rd.  5  cd.  ft.  9 

7.  i  cu.  ft.;  19bu.2 
I  pk.  7  qt.  1  pt. 
f  92r.  9q.  21  s. 

8.  \  130  A.  150  p. 
I    30  vd. 


3. 
4. 


[One,  79  oz.  14 
o  pwt. ;  each  of 

J     the  others,  39 
[    oz.  17  pwt. 

10.  6  mi.  176  rd.  0  vd. 

n  ft. 

,  ,      f  577  gal.  3  qt. 
11*    1 17  d.  4  h.  44  m. 
12.  1  cwt.  15  lb. 

Art.  407. 

1.  1000.0001 

2.  15.000015 
24.166 

f  92.9907 
\  .999999 

5.  .10816 

6.  U. 

7.  .176 

8.  10. 
9    1  ^ 

10.'  312.006 

11.  .000042 

12.  7*. 

13.  114.3055 

14.  35.375  gal. 

15.  .74375 

Art.  433. 

1.  476.85  vd. ; 

$13.75; 
9.36  bu. 

2.  .27  + 

3.  30. 

4.  35. 

5.  $8.62* 

6.  $51.45;  $42.87i 

7.  $393.86| 

8.  $8. 

9.  81415.62| 

10.  81102.67 

11.  86.27 

12.  $923.29 

13.  $18.96 

14.  81605.54 

15.  82111.11 


3 


QAlO 
IfeTl 


«— 


COWPERTHWAIT  &  CO.'S  EDUCATIONAL  SERIES. 


Hagar's  Mathematical  Series. 


Primary  Lessons  in  Numbers. 
Elementary  Arithmetic. 
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Greene's  Grammars  &  Language  Series. 


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3 __ 


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COWPERTHWAIT  &  CO.'S  EDUCATIONAL  SERIES. 


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•  ♦ « 1 

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Monroes   First   Reader. 
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These  Books  form  two  complete  Series,  adapted  to  the  different  grades 
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Primary  Lessons  in  Numbers]  FOR  TEACHERS. 

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Common  School  Arithmetic.  |  Key  to  Com.  School  Arithmetic. 
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